Computational approach in obtaining analytic solutions of a generalized nonlinear breaking soliton equation with applications in engineering and physics

Abstract

Higher-order nonlinear wave models have been a source of attraction to a huge number of researchers in recent times as a result of their significance in mathematical physics, other nonlinear sciences as well as engineering. In consequence, we outline in this paper the analytical studies entrenched on a generalized structure of a nonlinear breaking soliton equation with higher-order nonlinearity in four variables which have applications in science as well as engineering. Lie group theory is utilized to generate an 11-dimensional Lie algebra associated with the equation under consideration and in addition one parametric group of transformations related to the algebra is calculated. Besides, the technique is further invoked in performing reductions of the various subalgebras of the understudy model. Moreover, in conjunction with the theory, the direct integration technique is engaged to secure an analytic solution of the equation and as a result, a general analytic solution with regard to the second-kind elliptic-integral function is furnished. Moreover, we engage the novel simplest equation technique to gain more general solutions to the equation. In consequence, solitonic solutions comprising periodic, dark-bright, topological kink as well as singular solutions are achieved. We complemented that by exhibiting the dynamics of the secured solutions with the aid of graphical representations. In conclusion, we calculate conserved quantities associated with the aforementioned equation by invoking the well-celebrated classical Noether theorem technique.

Keywords:

• Generalized nonlinear breaking soliton equation with higher-order nonlinearity in four variables
• Lie group theory
• analytic solutions
• simplest equation approach
• conserved quantities

1. Introduction

It has been understood that nonlinear phenomena to a large extent possess an expansive scale in diverse arms of applications of mathematical physics as well as engineering. Nonlinearity is such an enthralling concept of the universe that countless scientists have contemplated nonlinear science virtually as a highly essential edge to aid the cardinal knowledge of the earth. Mathematical modelling of phenomena which are intricate and also change with time is dependent intensely on the study of a distinctive class of nonlinear ordinary differential equations (NLODEs) as well as nonlinear partial differential equations (NLPDEs). For instance, the femto-second optical pulses embedded in a double-refractive optical fibre, modelled into an NLPDE, were investigated in [1], the vector bright solitons, alongside their various interaction attribute related to the coupled Fokas–Lenells system were also studied. Recently, the authors in [2] examined a (3 + 1)-D nonlinear generalized type of potential Yu–Toda–Sasa–Fukuyama model existent in physics alongside engineering. These models have been introduced in highly unalike fields of study which range from the physical along with natural sciences, neural networks, infectious disease epidemiology, population ecology to economics, plasma physics, elasticity, optical fibres and fluid mechanics alongside solid-state physics. See more in [3–18].

Sophus Lie (1842–1899) with his quintessential work on Lie Algebras [19,20] which is essentially a unified approach for the treatment of a wide class of differential equations (DEs). With the inspiration of Galois theory, Sophus Lie, a Norwegian mathematician, established symmetry methods and demonstrated that many of the known ad-hoc methods of integration of DEs could be obtained in a systematic manner. The approach has evolved into a helpful tool for solving DEs, classifying them and preserving the solution set. In addition, we state that Lie group analysis furnishes a universal and robust approach through which tough nonlinear differential equations can be solved, once the symmetries are available which happens in most cases. This makes the technique stand out and performs much better than any other method.

Furthermore, it has been observed that conservation laws are established and entrenched natural laws that have been studied by many researchers in various scientific fields. Conservation laws that are commonly used in this context include conservation of linear momentum in an isolated system, conservation of electric charge, conservation of energy, conservation of mechanical energy in the absence of dissipative forces and many others. Conservation laws are deliberated to be basic laws of nature, with extensive application in physics and numerous other fields. Some of the importance of conservation laws include [21]:

1. the stability analysis and the global behaviour of solutions,

2. the development of numerical methods and provide an essential starting point for finding potential variables and nonlocally related systems,

3. the investigation of integrability and linearization mappings.

Closed-form solitary wave solutions provide us with better information about physical phenomena. Hence, substantial efforts have been devoted by several physical scientists as well as mathematicians to establish robust techniques in securing analytical along with numerical outcomes for NLPDEs.

Stating a few of the established techniques, we have extended homoclinic-test approach [22], generalized unified technique [23], tanh–coth technique [24], Bäcklund transformation approach [25], Cole–Hopf transformation approach [26], Painlevé expansion approach [27], Adomian decomposition method [28], mapping, as well as extended mapping approach [29], homotopy perturbation approach [30], rational expansion method [31], simplest as well as extended simplest equation [32], F-expansion approach [33], Hirota technique [34], Lie point symmetry analysis [19,20], the $\left(\frac{G^{\mathrm{\prime }}}{G}\right)$-expansion approach [35], Darboux transformation technique [36], sine-Gordon equation expansion approach [37], exponential function technique [38], Kudryashov's technique [39], tanh-function method [40], bifurcation approach [41] and the like.

Nonetheless, nonlinear task gives rise to some basic defiances that are readily problematic to check, due to nonlinear traits of the system distinctly transforming as a result of a negligible switch of valid-time-inclusive parameters. Consequently, the situation becomes more bewildering, which demands a paramount solution.

The breaking soliton system (BSS)[42] is expressed as (1) $\begin{array}{rl}& p_t+\alpha p_{\mathrm{xxy}}+4\mathrm{\alpha p}{q}_x+4\alpha p_{x}q=0,\end{array}$(1) (2) $\begin{array}{rl}& p_y-q_x=0,\end{array}$(2) where constant parameter $\alpha \ne 0$ was first established by Bogoyavlensky [42]. BSS (1(1) $\begin{array}{rl}& p_t+\alpha p_{\mathrm{xxy}}+4\mathrm{\alpha p}{q}_x+4\alpha p_{x}q=0,\end{array}$(1) ) portrays the existence of interaction of a long wave in regard to x-axis alongside a Riemann-wave propagating directly toward the y-axis. This fact was first described in 1977 by Degasperis and Calogero. Li et al. [43] engaged in an infinitesimal version of the dressing technique to construct diverse symmetries for BSS via Lie algebra. Moreover in [44], Zheng together with Zhang prepared some transformations and also went ahead to extend the generalized projective Riccati equation technique initiated in [45]. Besides the authors applied these approaches to BSS (1(1) $\begin{array}{rl}& p_t+\alpha p_{\mathrm{xxy}}+4\mathrm{\alpha p}{q}_x+4\alpha p_{x}q=0,\end{array}$(1) ) which made them recover six types of new distinctive exact solution-like results. Painlevé property, as well as the secured dromion-like structures for system (1(1) $\begin{array}{rl}& p_t+\alpha p_{\mathrm{xxy}}+4\mathrm{\alpha p}{q}_x+4\alpha p_{x}q=0,\end{array}$(1) ), was proven by Lakshmanan together with Radaha [46]. Furthermore, a general variable separation solution of BSS (1(1) $\begin{array}{rl}& p_t+\alpha p_{\mathrm{xxy}}+4\mathrm{\alpha p}{q}_x+4\alpha p_{x}q=0,\end{array}$(1) ) was achieved by Zhang et al. [47]. Darvishi and Najafi [48] compute some explicit closed-form solutions of (1(1) $\begin{array}{rl}& p_t+\alpha p_{\mathrm{xxy}}+4\mathrm{\alpha p}{q}_x+4\alpha p_{x}q=0,\end{array}$(1) ) through the concept of a three-wave-solution approach via Maple software. In [49], the authors found the analytic breather cross-kink solution for BSS (1(1) $\begin{array}{rl}& p_t+\alpha p_{\mathrm{xxy}}+4\mathrm{\alpha p}{q}_x+4\alpha p_{x}q=0,\end{array}$(1) ) by utilizing Hirota bilinear technique. In the same vein, Chen et al. [50] invoke the $\left(G^{\mathrm{\prime }}/G\right)$-expansion technique to gain new general but non-travelling wave solutions for the system under investigation. Chaotic solitons, in conjunction with various annihilation solitons, were secured via the improved mapping technique in [51]. Moreso, periodic-type soliton, breather-type solitonic outcome and two-solitonic solutions alongside three-soliton solutions concerning BSS (1(1) $\begin{array}{rl}& p_t+\alpha p_{\mathrm{xxy}}+4\mathrm{\alpha p}{q}_x+4\alpha p_{x}q=0,\end{array}$(1) ) were gained by utilizing the extended three-wave approach in [52]. In [53], multi-soliton solutions via the Wronskian approach as well as Hirota bilinear technique were invoked, as a result, double Wronskian solution and N-soliton solutions were gained by the authors. Li in [54] derived doubly periodic wave solutions along with diverse of the breather-type solitary-wave solutions for system (1(1) $\begin{array}{rl}& p_t+\alpha p_{\mathrm{xxy}}+4\mathrm{\alpha p}{q}_x+4\alpha p_{x}q=0,\end{array}$(1) ) by virtue of generalized three-wave approach with Hirota bilinear technique. Various structures of analytical solutions of BSS (1(1) $\begin{array}{rl}& p_t+\alpha p_{\mathrm{xxy}}+4\mathrm{\alpha p}{q}_x+4\alpha p_{x}q=0,\end{array}$(1) ) were achieved through the Darboux transformation along with mapping techniques in [55]. Hossein et al. in [56] employed the particular auxiliary function alongside Hirota bilinear approach to achieve solitary waves together with rogue wave solutions for BSS (1(1) $\begin{array}{rl}& p_t+\alpha p_{\mathrm{xxy}}+4\mathrm{\alpha p}{q}_x+4\alpha p_{x}q=0,\end{array}$(1) ). Curved solitons, dromion, lumps and breathers together with ring solitons solutions of BSS (1(1) $\begin{array}{rl}& p_t+\alpha p_{\mathrm{xxy}}+4\mathrm{\alpha p}{q}_x+4\alpha p_{x}q=0,\end{array}$(1) ) via Bäcklund transformations were achieved by Ruan in [57]. In [58], the authors on the basis of the singular manifold technique attained the periodic wave results of BSS (1(1) $\begin{array}{rl}& p_t+\alpha p_{\mathrm{xxy}}+4\mathrm{\alpha p}{q}_x+4\alpha p_{x}q=0,\end{array}$(1) ) with regard to Jacobi elliptic functions. Additionally, the similarity transformation approach based on the Lie group technique was investigated in constructing analytic solutions of BSS (1(1) $\begin{array}{rl}& p_t+\alpha p_{\mathrm{xxy}}+4\mathrm{\alpha p}{q}_x+4\alpha p_{x}q=0,\end{array}$(1) ). Besides, in [59], the authors gained quadratic, periodic, multi-solitons with asymptotic solutions to BSS (1(1) $\begin{array}{rl}& p_t+\alpha p_{\mathrm{xxy}}+4\mathrm{\alpha p}{q}_x+4\alpha p_{x}q=0,\end{array}$(1) ).

Two-dimensional breaking soliton equation [60] is stated as (3) $u_{\mathrm{xt}}-4u_x{u}_{\mathrm{xy}}-2u_{\mathrm{xx}}{u}_y-u_{\mathrm{xxxy}}=0.$(3) Diverse families of solutions have been achieved in the literature for model (3(3) $u_{\mathrm{xt}}-4u_x{u}_{\mathrm{xy}}-2u_{\mathrm{xx}}{u}_y-u_{\mathrm{xxxy}}=0.$(3) ) existing in dimensions of two [60–62] that delineate the interconnections that are existing between the longwave alongside Riemann wave propagating respectively along x with y-axes.

Nonlinear three-dimensional breaking soliton model (3D BSM) [63] that reads (4) $\begin{array}{rl}& u_{\mathrm{xxxy}}+u_{\mathrm{xxxz}}-u_{\mathrm{xt}}+4u_x{u}_{\mathrm{xy}}+4u_x{u}_{\mathrm{xz}}\\ & +2u_y{u}_{\mathrm{xx}}+2u_z{u}_{\mathrm{xx}}=0,\end{array}$(4) with $u=u(t,x,y,z):{\mathbb{R}}_t\times {\mathbb{R}}_x\times {\mathbb{R}}_y\times {\mathbb{R}}_z\to \mathbb{R}$.

Nonlinear 3D BSM (4(4) $\begin{array}{rl}& u_{\mathrm{xxxy}}+u_{\mathrm{xxxz}}-u_{\mathrm{xt}}+4u_x{u}_{\mathrm{xy}}+4u_x{u}_{\mathrm{xz}}\\ & +2u_y{u}_{\mathrm{xx}}+2u_z{u}_{\mathrm{xx}}=0,\end{array}$(4) ) has been at the forefront of study by a good number of researchers. Demiray et al. [63] secured an approximate solution of the equation by invoking the perturbed homotopy technique. Besides, the authors formed a chart that comprises numerical results and concluded by presenting the graphics of the secured solution. In [64], some classical solutions of nonlinear 3D BSM (4(4) $\begin{array}{rl}& u_{\mathrm{xxxy}}+u_{\mathrm{xxxz}}-u_{\mathrm{xt}}+4u_x{u}_{\mathrm{xy}}+4u_x{u}_{\mathrm{xz}}\\ & +2u_y{u}_{\mathrm{xx}}+2u_z{u}_{\mathrm{xx}}=0,\end{array}$(4) ) were gained after Hirota's bilinear structure was formed. Also, the authors revealed that the three-wave technique is easy to achieve solutions of nonlinear partial differentiation. Residual symmetries of (4(4) $\begin{array}{rl}& u_{\mathrm{xxxy}}+u_{\mathrm{xxxz}}-u_{\mathrm{xt}}+4u_x{u}_{\mathrm{xy}}+4u_x{u}_{\mathrm{xz}}\\ & +2u_y{u}_{\mathrm{xx}}+2u_z{u}_{\mathrm{xx}}=0,\end{array}$(4) ) were obtained in [65]. The authors went further to prove that Equation (4(4) $\begin{array}{rl}& u_{\mathrm{xxxy}}+u_{\mathrm{xxxz}}-u_{\mathrm{xt}}+4u_x{u}_{\mathrm{xy}}+4u_x{u}_{\mathrm{xz}}\\ & +2u_y{u}_{\mathrm{xx}}+2u_z{u}_{\mathrm{xx}}=0,\end{array}$(4) ) is integrable due to it possessing a Riccati expansion persistently. Tariq et al. [66] secured some travelling wave solutions of (4(4) $\begin{array}{rl}& u_{\mathrm{xxxy}}+u_{\mathrm{xxxz}}-u_{\mathrm{xt}}+4u_x{u}_{\mathrm{xy}}+4u_x{u}_{\mathrm{xz}}\\ & +2u_y{u}_{\mathrm{xx}}+2u_z{u}_{\mathrm{xx}}=0,\end{array}$(4) ) where they observed that the outcome held key importance in explicating physical situations. Furthermore, Pallavi et al. [67] gained some analytical solutions of a version of (4(4) $\begin{array}{rl}& u_{\mathrm{xxxy}}+u_{\mathrm{xxxz}}-u_{\mathrm{xt}}+4u_x{u}_{\mathrm{xy}}+4u_x{u}_{\mathrm{xz}}\\ & +2u_y{u}_{\mathrm{xx}}+2u_z{u}_{\mathrm{xx}}=0,\end{array}$(4) ) via the engagement of extended-exp$(-\varphi (\xi \left)\right)$ -expansion technique. Al-Amr [68] obtained closed-form solutions of a different version of (4(4) $\begin{array}{rl}& u_{\mathrm{xxxy}}+u_{\mathrm{xxxz}}-u_{\mathrm{xt}}+4u_x{u}_{\mathrm{xy}}+4u_x{u}_{\mathrm{xz}}\\ & +2u_y{u}_{\mathrm{xx}}+2u_z{u}_{\mathrm{xx}}=0,\end{array}$(4) ) when u is not depending on t through the use of a modified simple equation technique. In [69], Xu et al. investigated the bidirectional solitons, integrability characteristics as well as localized solutions of a version of (4(4) $\begin{array}{rl}& u_{\mathrm{xxxy}}+u_{\mathrm{xxxz}}-u_{\mathrm{xt}}+4u_x{u}_{\mathrm{xy}}+4u_x{u}_{\mathrm{xz}}\\ & +2u_y{u}_{\mathrm{xx}}+2u_z{u}_{\mathrm{xx}}=0,\end{array}$(4) ). The authors performed a singularity manifold analysis of the equation and achieved a new model which existed in the sense of Painlevé condition. Additionally, localized excitations, periodic, lump and solitary waves, as well as multi-dromion solutions, were gained. Having explicated various research works carried out on 3D BSM (4(4) $\begin{array}{rl}& u_{\mathrm{xxxy}}+u_{\mathrm{xxxz}}-u_{\mathrm{xt}}+4u_x{u}_{\mathrm{xy}}+4u_x{u}_{\mathrm{xz}}\\ & +2u_y{u}_{\mathrm{xx}}+2u_z{u}_{\mathrm{xx}}=0,\end{array}$(4) ), we go ahead to generalize the equation in the subsequent part of the work. For the first time, we carry out a detailed investigation in performing a Lie group analysis on the generalized version. Besides, for the very first time, we secure various Noether's symmetry of the generalized version and utilize them to obtain abundant conserved vectors of the equation and all these portray the novelty of our research work.

1.1. Governing equation

In this research, we contemplate investigating a generalized nonlinear (3 + 1)-D breaking soliton equation (3D-genBSMe) stated as (5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) with parameters θ, α, β, γ regarded as real value constants which are nonzero. We recover the generalized version (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) from the earlier-presented model (4(4) $\begin{array}{rl}& u_{\mathrm{xxxy}}+u_{\mathrm{xxxz}}-u_{\mathrm{xt}}+4u_x{u}_{\mathrm{xy}}+4u_x{u}_{\mathrm{xz}}\\ & +2u_y{u}_{\mathrm{xx}}+2u_z{u}_{\mathrm{xx}}=0,\end{array}$(4) ), if we assume $\theta =2$, and $\alpha =\text{β}=\gamma =1$.

Having earlier explicated the originality/novelty of the work in this paper, the research paper is thus arranged in the following structure, viz. Section 1 furnishes the introduction alongside the literature of the understudy model. In Section 2, the implementation of Lie group analysis of 3D-genBSMe (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) is achieved, and we compute various groups of transformations corresponding to the obtained results. Besides, Lie point symmetries of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) are implored in performing reductions on the equation. Section 3 employs the simplest equation technique to gain exact solutions of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ). The discussion of results alongside their pictorial representations is done in Section 4. Moreover, we present in Section 5, the conservation laws ascribed to the underlying model by utilizing the conserved-vector theorem by Noether. Finally, concluding remarks are presented afterwards.

2. Lie group analysis of (eqn5)

One first presents the step-by-step way of computing the Lie point symmetries of 3D-genBSMe (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) which at the end will be used in the construction of exact solutions of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ).

2.1. Lie point symmetries of 3D-genBSMe (5)

We examine the Lie algebra with an infinitesimal dimension that is spanned via the vector fields explicated as $\begin{array}{rl}\mathcal{P}& ={\xi }^1(t,x,y,z,u)\frac{\mathrm{\partial }}{\mathrm{\partial }t}+{\xi }^2(t,x,y,z,u)\frac{\mathrm{\partial }}{\mathrm{\partial }x}\\ & +{\xi }^3(t,x,y,z,u)\frac{\mathrm{\partial }}{\mathrm{\partial }y}\\ & +{\xi }^4(t,x,y,z,u)\frac{\mathrm{\partial }}{\mathrm{\partial }z}+\eta (t,x,y,z,u)\frac{\mathrm{\partial }}{\mathrm{\partial }u},\end{array}$ with the vector field giving the required symmetries of 3D-genBSMe (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) and $\mathcal{P}$ satisfying the invariance criteria (6) $\begin{array}{rl}& {\mathcal{P}}^{\left[4\right]}[\alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}\\ & +2\theta u_x{u}_{\mathrm{xz}}+\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}]=0,\end{array}$(6) whenever $-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}+\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}+\alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}=0$, where ${\mathcal{P}}^{\left[4\right]}$ connotes the extension of $\mathcal{P}$. Corresponding formula for ${\mathcal{P}}^{\left[4\right]}$ reads (7) $\begin{array}{rl}{\mathcal{P}}^{\left[4\right]}& =\mathcal{P}+{\eta }^t{\mathrm{\partial }}_{u_t}+{\eta }^x{\mathrm{\partial }}_{u_x}+{\eta }^y{\mathrm{\partial }}_{u_y}+{\eta }^z{\mathrm{\partial }}_{u_z}\\ & +{\eta }^{\mathrm{tx}}{\mathrm{\partial }}_{u_{\mathrm{tx}}}+{\eta }^{\mathrm{xx}}{\mathrm{\partial }}_{u_{\mathrm{xx}}}\\ & +{\eta }^{\mathrm{xy}}{\mathrm{\partial }}_{u_{\mathrm{xy}}}+{\eta }^{\mathrm{xz}}{\mathrm{\partial }}_{u_{\mathrm{xz}}}+{\eta }^{\mathrm{xxxy}}{\mathrm{\partial }}_{u_{\mathrm{xxxy}}}+{\eta }^{\mathrm{xxxz}}{\mathrm{\partial }}_{u_{\mathrm{xxxz}}}.\end{array}$(7) The application of ${\mathcal{P}}^{\left[4\right]}$ to model (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) occasions the invariant conditions expressed as (8) $\begin{array}{rl}& -\gamma {\eta }^{\mathrm{xt}}+2\theta ({\eta }^x{u}_{\mathrm{xy}}+u_x{\eta }^{\mathrm{xy}}+{\eta }^x{u}_{\mathrm{xz}}+u_x{\eta }^{\mathrm{xz}})\\ & +\theta ({\eta }^{\mathrm{xx}}{u}_y+u_{\mathrm{xx}}{\eta }^y+{\eta }^{\mathrm{xx}}{u}_z+u_{\mathrm{xx}}{\eta }^z)\\ & +\alpha {\eta }^{\mathrm{xxxy}}+\text{β}{\eta }^{\mathrm{xxxz}}=0,\end{array}$(8) with the coefficients of ${\mathcal{P}}^{\left[4\right]}$, that is, ${\eta }^x$, ${\eta }^y$, ${\eta }^z$, ${\eta }^{\mathrm{xt}}$, ${\eta }^{\mathrm{xx}}$, ${\eta }^{\mathrm{xy}}$, ${\eta }^{\mathrm{xz}}$, ${\eta }^{\mathrm{xxxy}}$ as well as ${\eta }^{\mathrm{xxxz}}$ given as (9) $\begin{array}{rl}{\eta }^t& =D_t\left(\eta \right)-u_t{D}_t\left({\xi }^1\right)-u_x{D}_t\left({\xi }^2\right)\\ & -u_y{D}_t\left({\xi }^3\right)-u_z{D}_t\left({\xi }^4\right),\\ {\eta }^x& =D_x\left(\eta \right)-u_t{D}_x\left({\xi }^1\right)-u_x{D}_x\left({\xi }^2\right)\\ & -u_y{D}_x\left({\xi }^3\right)-u_z{D}_x\left({\xi }^4\right),\\ {\eta }^y& =D_y\left(\eta \right)-u_t{D}_y\left({\xi }^1\right)-u_x{D}_y\left({\xi }^2\right)\\ & -u_y{D}_y\left({\xi }^3\right)-u_z{D}_y\left({\xi }^4\right),\\ {\eta }^z& =D_z\left(\eta \right)-u_t{D}_z\left({\xi }^1\right)-u_x{D}_z\left({\xi }^2\right)\\ & -u_y{D}_z\left({\xi }^3\right)-u_z{D}_z\left({\xi }^4\right),\\ {\eta }^{\mathrm{tx}}& =D_x\left({\eta }^t\right)-u_{\mathrm{tt}}{D}_x\left({\xi }^1\right)-u_{\mathrm{tx}}{D}_x\left({\xi }^2\right)\\ & -u_{\mathrm{yt}}{D}_x\left({\xi }^3\right)-u_{\mathrm{zt}}{D}_x\left({\xi }^4\right),\\ {\eta }^{\mathrm{xx}}& =D_x\left({\eta }^x\right)-u_{\mathrm{tx}}{D}_x\left({\xi }^1\right)-u_{\mathrm{xx}}{D}_x\left({\xi }^2\right)\\ & -u_{\mathrm{xy}}{D}_x\left({\xi }^3\right)-u_{\mathrm{xz}}{D}_x\left({\xi }^4\right),\\ {\eta }^{\mathrm{xy}}& =D_x\left({\eta }^y\right)-u_{\mathrm{yt}}{D}_x\left({\xi }^1\right)-u_{\mathrm{xy}}{D}_x\left({\xi }^2\right)\\ & -u_{\mathrm{yy}}{D}_x\left({\xi }^3\right)-u_{\mathrm{zy}}{D}_x\left({\xi }^4\right),\\ {\eta }^{\mathrm{xz}}& =D_x\left({\eta }^z\right)-u_{\mathrm{zt}}{D}_x\left({\xi }^1\right)-u_{\mathrm{xz}}{D}_x\left({\xi }^2\right)\\ & -u_{\mathrm{zy}}{D}_x\left({\xi }^3\right)-u_{\mathrm{zz}}{D}_x\left({\xi }^4\right),\\ {\eta }^{\mathrm{xxxy}}& =D_x\left({\eta }^{\mathrm{xxy}}\right)-u_{\mathrm{xxyt}}{D}_x\left({\xi }^1\right)-u_{\mathrm{xxxy}}{D}_x\left({\xi }^2\right)\\ & -u_{\mathrm{xxyy}}{D}_x\left({\xi }^3\right)-u_{\mathrm{zyxx}}{D}_x\left({\xi }^4\right),\\ {\eta }^{\mathrm{xxxz}}& =D_x\left({\eta }^{\mathrm{xxz}}\right)-u_{\mathrm{txxz}}{D}_x\left({\xi }^1\right)-u_{\mathrm{xxxz}}{D}_x\left({\xi }^2\right)\\ & -u_{\mathrm{zyxx}}{D}_x\left({\xi }^3\right)-u_{\mathrm{zzxx}}{D}_x\left({\xi }^4\right).\end{array}$(9) (See the detailed expansion of the coefficients (9(9) $\begin{array}{rl}{\eta }^t& =D_t\left(\eta \right)-u_t{D}_t\left({\xi }^1\right)-u_x{D}_t\left({\xi }^2\right)\\ & -u_y{D}_t\left({\xi }^3\right)-u_z{D}_t\left({\xi }^4\right),\\ {\eta }^x& =D_x\left(\eta \right)-u_t{D}_x\left({\xi }^1\right)-u_x{D}_x\left({\xi }^2\right)\\ & -u_y{D}_x\left({\xi }^3\right)-u_z{D}_x\left({\xi }^4\right),\\ {\eta }^y& =D_y\left(\eta \right)-u_t{D}_y\left({\xi }^1\right)-u_x{D}_y\left({\xi }^2\right)\\ & -u_y{D}_y\left({\xi }^3\right)-u_z{D}_y\left({\xi }^4\right),\\ {\eta }^z& =D_z\left(\eta \right)-u_t{D}_z\left({\xi }^1\right)-u_x{D}_z\left({\xi }^2\right)\\ & -u_y{D}_z\left({\xi }^3\right)-u_z{D}_z\left({\xi }^4\right),\\ {\eta }^{\mathrm{tx}}& =D_x\left({\eta }^t\right)-u_{\mathrm{tt}}{D}_x\left({\xi }^1\right)-u_{\mathrm{tx}}{D}_x\left({\xi }^2\right)\\ & -u_{\mathrm{yt}}{D}_x\left({\xi }^3\right)-u_{\mathrm{zt}}{D}_x\left({\xi }^4\right),\\ {\eta }^{\mathrm{xx}}& =D_x\left({\eta }^x\right)-u_{\mathrm{tx}}{D}_x\left({\xi }^1\right)-u_{\mathrm{xx}}{D}_x\left({\xi }^2\right)\\ & -u_{\mathrm{xy}}{D}_x\left({\xi }^3\right)-u_{\mathrm{xz}}{D}_x\left({\xi }^4\right),\\ {\eta }^{\mathrm{xy}}& =D_x\left({\eta }^y\right)-u_{\mathrm{yt}}{D}_x\left({\xi }^1\right)-u_{\mathrm{xy}}{D}_x\left({\xi }^2\right)\\ & -u_{\mathrm{yy}}{D}_x\left({\xi }^3\right)-u_{\mathrm{zy}}{D}_x\left({\xi }^4\right),\\ {\eta }^{\mathrm{xz}}& =D_x\left({\eta }^z\right)-u_{\mathrm{zt}}{D}_x\left({\xi }^1\right)-u_{\mathrm{xz}}{D}_x\left({\xi }^2\right)\\ & -u_{\mathrm{zy}}{D}_x\left({\xi }^3\right)-u_{\mathrm{zz}}{D}_x\left({\xi }^4\right),\\ {\eta }^{\mathrm{xxxy}}& =D_x\left({\eta }^{\mathrm{xxy}}\right)-u_{\mathrm{xxyt}}{D}_x\left({\xi }^1\right)-u_{\mathrm{xxxy}}{D}_x\left({\xi }^2\right)\\ & -u_{\mathrm{xxyy}}{D}_x\left({\xi }^3\right)-u_{\mathrm{zyxx}}{D}_x\left({\xi }^4\right),\\ {\eta }^{\mathrm{xxxz}}& =D_x\left({\eta }^{\mathrm{xxz}}\right)-u_{\mathrm{txxz}}{D}_x\left({\xi }^1\right)-u_{\mathrm{xxxz}}{D}_x\left({\xi }^2\right)\\ & -u_{\mathrm{zyxx}}{D}_x\left({\xi }^3\right)-u_{\mathrm{zzxx}}{D}_x\left({\xi }^4\right).\end{array}$(9) ) in the appendix.) We have the total derivatives ($D_t$, $D_x$, $D_y$, $D_z$) expressed as [20] (10) $\begin{array}{rl}& D_t=\frac{\mathrm{\partial }}{\mathrm{\partial }t}+u_t\frac{\mathrm{\partial }}{\mathrm{\partial }u}+u_{\mathrm{tt}}\frac{\mathrm{\partial }}{\mathrm{\partial }{u}_t}+u_{\mathrm{yt}}\frac{\mathrm{\partial }}{\mathrm{\partial }{u}_y}+\cdots .\\ & D_x=\frac{\mathrm{\partial }}{\mathrm{\partial }x}+u_x\frac{\mathrm{\partial }}{\mathrm{\partial }u}+u_{\mathrm{xx}}\frac{\mathrm{\partial }}{\mathrm{\partial }{u}_x}+u_{\mathrm{xt}}\frac{\mathrm{\partial }}{\mathrm{\partial }{u}_t}+\cdots ,\\ & D_y=\frac{\mathrm{\partial }}{\mathrm{\partial }y}+u_y\frac{\mathrm{\partial }}{\mathrm{\partial }u}+u_{\mathrm{yy}}\frac{\mathrm{\partial }}{\mathrm{\partial }{u}_y}+u_{\mathrm{yt}}\frac{\mathrm{\partial }}{\mathrm{\partial }{u}_t}+\cdots ,\\ & D_z=\frac{\mathrm{\partial }}{\mathrm{\partial }z}+u_z\frac{\mathrm{\partial }}{\mathrm{\partial }u}+u_{\mathrm{zz}}\frac{\mathrm{\partial }}{\mathrm{\partial }{u}_z}+u_{\mathrm{zt}}\frac{\mathrm{\partial }}{\mathrm{\partial }{u}_t}+\cdots .\end{array}$(10) Applying Equation (9(9) $\begin{array}{rl}{\eta }^t& =D_t\left(\eta \right)-u_t{D}_t\left({\xi }^1\right)-u_x{D}_t\left({\xi }^2\right)\\ & -u_y{D}_t\left({\xi }^3\right)-u_z{D}_t\left({\xi }^4\right),\\ {\eta }^x& =D_x\left(\eta \right)-u_t{D}_x\left({\xi }^1\right)-u_x{D}_x\left({\xi }^2\right)\\ & -u_y{D}_x\left({\xi }^3\right)-u_z{D}_x\left({\xi }^4\right),\\ {\eta }^y& =D_y\left(\eta \right)-u_t{D}_y\left({\xi }^1\right)-u_x{D}_y\left({\xi }^2\right)\\ & -u_y{D}_y\left({\xi }^3\right)-u_z{D}_y\left({\xi }^4\right),\\ {\eta }^z& =D_z\left(\eta \right)-u_t{D}_z\left({\xi }^1\right)-u_x{D}_z\left({\xi }^2\right)\\ & -u_y{D}_z\left({\xi }^3\right)-u_z{D}_z\left({\xi }^4\right),\\ {\eta }^{\mathrm{tx}}& =D_x\left({\eta }^t\right)-u_{\mathrm{tt}}{D}_x\left({\xi }^1\right)-u_{\mathrm{tx}}{D}_x\left({\xi }^2\right)\\ & -u_{\mathrm{yt}}{D}_x\left({\xi }^3\right)-u_{\mathrm{zt}}{D}_x\left({\xi }^4\right),\\ {\eta }^{\mathrm{xx}}& =D_x\left({\eta }^x\right)-u_{\mathrm{tx}}{D}_x\left({\xi }^1\right)-u_{\mathrm{xx}}{D}_x\left({\xi }^2\right)\\ & -u_{\mathrm{xy}}{D}_x\left({\xi }^3\right)-u_{\mathrm{xz}}{D}_x\left({\xi }^4\right),\\ {\eta }^{\mathrm{xy}}& =D_x\left({\eta }^y\right)-u_{\mathrm{yt}}{D}_x\left({\xi }^1\right)-u_{\mathrm{xy}}{D}_x\left({\xi }^2\right)\\ & -u_{\mathrm{yy}}{D}_x\left({\xi }^3\right)-u_{\mathrm{zy}}{D}_x\left({\xi }^4\right),\\ {\eta }^{\mathrm{xz}}& =D_x\left({\eta }^z\right)-u_{\mathrm{zt}}{D}_x\left({\xi }^1\right)-u_{\mathrm{xz}}{D}_x\left({\xi }^2\right)\\ & -u_{\mathrm{zy}}{D}_x\left({\xi }^3\right)-u_{\mathrm{zz}}{D}_x\left({\xi }^4\right),\\ {\eta }^{\mathrm{xxxy}}& =D_x\left({\eta }^{\mathrm{xxy}}\right)-u_{\mathrm{xxyt}}{D}_x\left({\xi }^1\right)-u_{\mathrm{xxxy}}{D}_x\left({\xi }^2\right)\\ & -u_{\mathrm{xxyy}}{D}_x\left({\xi }^3\right)-u_{\mathrm{zyxx}}{D}_x\left({\xi }^4\right),\\ {\eta }^{\mathrm{xxxz}}& =D_x\left({\eta }^{\mathrm{xxz}}\right)-u_{\mathrm{txxz}}{D}_x\left({\xi }^1\right)-u_{\mathrm{xxxz}}{D}_x\left({\xi }^2\right)\\ & -u_{\mathrm{zyxx}}{D}_x\left({\xi }^3\right)-u_{\mathrm{zzxx}}{D}_x\left({\xi }^4\right).\end{array}$(9) ) in (8(8) $\begin{array}{rl}& -\gamma {\eta }^{\mathrm{xt}}+2\theta ({\eta }^x{u}_{\mathrm{xy}}+u_x{\eta }^{\mathrm{xy}}+{\eta }^x{u}_{\mathrm{xz}}+u_x{\eta }^{\mathrm{xz}})\\ & +\theta ({\eta }^{\mathrm{xx}}{u}_y+u_{\mathrm{xx}}{\eta }^y+{\eta }^{\mathrm{xx}}{u}_z+u_{\mathrm{xx}}{\eta }^z)\\ & +\alpha {\eta }^{\mathrm{xxxy}}+\text{β}{\eta }^{\mathrm{xxxz}}=0,\end{array}$(8) ), we equate the resulting differential coefficients of u together with its diverse derivatives to zero. Later, we solve the determining equations and secure system $\begin{array}{rl}& {\xi }_x^1=0,{\xi }_x^3=0,{\xi }_x^4=0,{\xi }_y^1=0,{\xi }_y^2=0,\\ & {\xi }_z^1=0,{\xi }_z^2=0,{\xi }_u^1=0,\\ & {\xi }_u^2=0,{\xi }_u^3=0,{\xi }_u^z=0,{\eta }_{\mathrm{tx}}=0,{\eta }_{\mathrm{ut}}=0,\\ & {\eta }_{\mathrm{xx}}=0,{\eta }_{\mathrm{xy}}=0,{\eta }_{\mathrm{xz}}=0,\\ & {\xi }_{\mathrm{zz}}^4=0,{\xi }_{\mathrm{zz}}^3=0,{\eta }_{\mathrm{uy}}=0,{\eta }_{\mathrm{ux}}=0,\\ & {\eta }_{\mathrm{yy}}-{\eta }_{\mathrm{zz}}=0,{\eta }_{\mathrm{yz}}+{\eta }_{\mathrm{zz}}=0,\\ & {\eta }_{\mathrm{uz}}=0{\eta }_{\mathrm{uu}}=0\gamma {\xi }_t^2+\theta ({\eta }_y+{\eta }_z)=0,\\ & \gamma {\xi }_t^3+2\theta {\eta }_x=0,\gamma {\xi }_t^4+2\theta {\eta }_x=0,\\ & \alpha {\xi }_y^4+\text{β}{\xi }_z^3=0,\alpha {\xi }_y^3-\alpha {\xi }_z^4+\alpha {\xi }_z^3+\text{β}{\xi }_z^3=0,\\ & \alpha {\xi }_x^2+\alpha {\eta }_u-\alpha {\xi }_z^3+\text{β}{\xi }_z^3=0,\\ & \alpha {\xi }_t^1+2\alpha {\eta }_u-\alpha {\xi }_z^4+2\text{β}{\xi }_z^3-\alpha {\xi }_z^3=0.\end{array}$ In solving the 31 system of equations, the result furnishes infinitesimal generators of one-parameter Lie symmetries of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) as $\begin{array}{rl}& {\xi }^1=(-2{\mathbf{c}}_2+2{\mathbf{c}}_3+{\mathbf{c}}_4)t+{\mathbf{c}}_8-\frac{\alpha }{\text{β}}{\mathbf{c}}_{3}t,\\ & {\xi }^2=-\frac{2\theta }{\gamma }{G}_3\left(t\right)-{\mathbf{c}}_{2}x-\frac{\alpha }{\text{β}}{\mathbf{c}}_{3}x+{\mathbf{c}}_{3}x+{\mathbf{c}}_7,\\ & {\xi }^3=-\frac{\alpha }{\text{β}}{\mathbf{c}}_{3}z-\frac{2\theta }{\gamma }{\mathbf{c}}_{1}t+\frac{\alpha }{\text{β}}{\mathbf{c}}_{3}y+{\mathbf{c}}_{3}y+{\mathbf{c}}_{4}y+{\mathbf{c}}_6,\\ & {\xi }^4=-\frac{2\theta }{\gamma }{\mathbf{c}}_{1}t+{\mathbf{c}}_{3}y+{\mathbf{c}}_{4}z+{\mathbf{c}}_5,\\ & \eta ={\mathbf{c}}_{1}x+{\mathbf{c}}_{2}u+(y+z)G_3^{\mathrm{\prime }}\left(t\right)+G_1\left(t\right)+G_2(t,z-y).\end{array}$ Hence, we secure 11-dimensional Lie algebra ascribed to infinitesimal symmetries of 3D-genBSMe (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) spanned via the vector fields, namely: (11) $\begin{array}{rl}{\mathcal{P}}_1& =\frac{\mathrm{\partial }}{\mathrm{\partial }t},{\mathcal{P}}_2=\frac{\mathrm{\partial }}{\mathrm{\partial }x},{\mathcal{P}}_3=\frac{\mathrm{\partial }}{\mathrm{\partial }y},\\ {\mathcal{P}}_4& =\frac{\mathrm{\partial }}{\mathrm{\partial }z},{\mathcal{P}}_5=G_1\left(t\right)\frac{\mathrm{\partial }}{\mathrm{\partial }u},{\mathcal{P}}_6=G(t,z-y)\frac{\mathrm{\partial }}{\mathrm{\partial }u},\\ {\mathcal{P}}_7& =u\frac{\mathrm{\partial }}{\mathrm{\partial }u}-2t\frac{\mathrm{\partial }}{\mathrm{\partial }t}-x\frac{\mathrm{\partial }}{\mathrm{\partial }x},{\mathcal{P}}_8=t\frac{\mathrm{\partial }}{\mathrm{\partial }t}+y\frac{\mathrm{\partial }}{\mathrm{\partial }y}+z\frac{\mathrm{\partial }}{\mathrm{\partial }z},\\ {\mathcal{P}}_9& =x\frac{\mathrm{\partial }}{\mathrm{\partial }u}-\frac{2\theta }{\gamma }t\frac{\mathrm{\partial }}{\mathrm{\partial }y}-\frac{2\theta }{\gamma }t\frac{\mathrm{\partial }}{\mathrm{\partial }z}\\ {\mathcal{P}}_{10}& =(y+z)G_2^{\mathrm{\prime }}\left(t\right)\frac{\mathrm{\partial }}{\mathrm{\partial }u}-\frac{2\theta }{\gamma }{G}_2\left(t\right)\frac{\mathrm{\partial }}{\mathrm{\partial }x},\\ {\mathcal{P}}_{11}& =\left(\frac{\alpha -2\text{β}}{\text{β}}\right)t\frac{\mathrm{\partial }}{\mathrm{\partial }t}+\left(\frac{\alpha -\text{β}}{\text{β}}\right)x\frac{\mathrm{\partial }}{\mathrm{\partial }x}\\ & -\left(\frac{\alpha (y-z)+\text{β}y}{\text{β}}\right)\frac{\mathrm{\partial }}{\mathrm{\partial }y}-y\frac{\mathrm{\partial }}{\mathrm{\partial }z}.\end{array}$(11) $G,G_1$ and $G_2$ are arbitrary functions.

2.2. Symmetry reductions and group invariants of (eqn5)

Here, we present the detailed calculations of group-invariant solutions of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) with the use of the Lie point symmetries given in (11(11) $\begin{array}{rl}{\mathcal{P}}_1& =\frac{\mathrm{\partial }}{\mathrm{\partial }t},{\mathcal{P}}_2=\frac{\mathrm{\partial }}{\mathrm{\partial }x},{\mathcal{P}}_3=\frac{\mathrm{\partial }}{\mathrm{\partial }y},\\ {\mathcal{P}}_4& =\frac{\mathrm{\partial }}{\mathrm{\partial }z},{\mathcal{P}}_5=G_1\left(t\right)\frac{\mathrm{\partial }}{\mathrm{\partial }u},{\mathcal{P}}_6=G(t,z-y)\frac{\mathrm{\partial }}{\mathrm{\partial }u},\\ {\mathcal{P}}_7& =u\frac{\mathrm{\partial }}{\mathrm{\partial }u}-2t\frac{\mathrm{\partial }}{\mathrm{\partial }t}-x\frac{\mathrm{\partial }}{\mathrm{\partial }x},{\mathcal{P}}_8=t\frac{\mathrm{\partial }}{\mathrm{\partial }t}+y\frac{\mathrm{\partial }}{\mathrm{\partial }y}+z\frac{\mathrm{\partial }}{\mathrm{\partial }z},\\ {\mathcal{P}}_9& =x\frac{\mathrm{\partial }}{\mathrm{\partial }u}-\frac{2\theta }{\gamma }t\frac{\mathrm{\partial }}{\mathrm{\partial }y}-\frac{2\theta }{\gamma }t\frac{\mathrm{\partial }}{\mathrm{\partial }z}\\ {\mathcal{P}}_{10}& =(y+z)G_2^{\mathrm{\prime }}\left(t\right)\frac{\mathrm{\partial }}{\mathrm{\partial }u}-\frac{2\theta }{\gamma }{G}_2\left(t\right)\frac{\mathrm{\partial }}{\mathrm{\partial }x},\\ {\mathcal{P}}_{11}& =\left(\frac{\alpha -2\text{β}}{\text{β}}\right)t\frac{\mathrm{\partial }}{\mathrm{\partial }t}+\left(\frac{\alpha -\text{β}}{\text{β}}\right)x\frac{\mathrm{\partial }}{\mathrm{\partial }x}\\ & -\left(\frac{\alpha (y-z)+\text{β}y}{\text{β}}\right)\frac{\mathrm{\partial }}{\mathrm{\partial }y}-y\frac{\mathrm{\partial }}{\mathrm{\partial }z}.\end{array}$(11) ).

2.2.1. Invariant solutions using operator ${\mathcal{P}}_1$

We engage operator ${\mathcal{P}}_1=\mathrm{\partial }/\mathrm{\partial t}$, where its Lagrangian system is expressed as (12) ${\displaystyle \frac{\mathrm{d}t}{1}}={\displaystyle \frac{\mathrm{d}x}{0}}={\displaystyle \frac{\mathrm{d}y}{0}}={\displaystyle \frac{\mathrm{d}z}{0}}={\displaystyle \frac{\mathrm{d}u}{0}}.$(12) Solution to the system gives function $u(t,x,y,z)=U(X,Y,Z)$, where $X=x$, $Y=y$ as well as $Z=z$. On inserting the function in Equation (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ), we achieve NLPDE (13) $\begin{array}{rl}& \theta U_Y{U}_{\mathrm{XX}}+\theta U_Z{U}_{\mathrm{XX}}+2\theta U_X{U}_{\mathrm{XY}}+2\theta U_X{U}_{\mathrm{XZ}}\\ & +\alpha U_{\mathrm{XXXY}}+\text{β}{U}_{\mathrm{XXXZ}}=0.\end{array}$(13) Invoking Lie theoretic approach on (13(13) $\begin{array}{rl}& \theta U_Y{U}_{\mathrm{XX}}+\theta U_Z{U}_{\mathrm{XX}}+2\theta U_X{U}_{\mathrm{XY}}+2\theta U_X{U}_{\mathrm{XZ}}\\ & +\alpha U_{\mathrm{XXXY}}+\text{β}{U}_{\mathrm{XXXZ}}=0.\end{array}$(13) ) supplies six symmetries: namely $\begin{array}{rl}{Q}_1& ={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Z}}+{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}},Q_2={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Y}}+{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}},Q_3={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }X}}+{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}},\\ Q_4& =Y{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Y}}+Z{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Z}}+{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}},Q_5=X{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }X}}+(1-U){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}},\\ Q_6& ={\displaystyle \frac{1}{\text{β}}}[\text{β}Y+\alpha (Y-Z)]{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Y}}\\ & +Y{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Z}}+(1+U-{\displaystyle \frac{\alpha }{\text{β}}}U){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}}.\end{array}$ We first invoke $Q_1$ and by solving the analogous Lagrangian system, one gains $U(X,Y,Z)=W(r,s)+Z$, where $r=X$ alongside $s=Y$. Substituting the achieved function in (13(13) $\begin{array}{rl}& \theta U_Y{U}_{\mathrm{XX}}+\theta U_Z{U}_{\mathrm{XX}}+2\theta U_X{U}_{\mathrm{XY}}+2\theta U_X{U}_{\mathrm{XZ}}\\ & +\alpha U_{\mathrm{XXXY}}+\text{β}{U}_{\mathrm{XXXZ}}=0.\end{array}$(13) ), we have (14) $\theta W_{\mathrm{rr}}+\theta W_s{W}_{\mathrm{rr}}+2\theta W_r{W}_{\mathrm{rs}}+\alpha W_{\mathrm{rrrs}}=0,$(14) which solves to give a topological kink solution of 3D-genBSMe (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) in this instance as (15) $\begin{array}{rl}& u(t,x,y,z)\\ & =z+{\displaystyle \frac{4\alpha A_2}{\theta }}\tanh (A_{2}x-{\displaystyle \frac{\theta }{4\alpha A_2}}y+A_1)+A_3\end{array}$(15) with constants $A_1$, $A_2$ and $A_3$ are arbitrary. We explore (14(14) $\theta W_{\mathrm{rr}}+\theta W_s{W}_{\mathrm{rr}}+2\theta W_r{W}_{\mathrm{rs}}+\alpha W_{\mathrm{rrrs}}=0,$(14) ) and then have $\begin{array}{rl}{G}_1& ={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }r}}+F_1\left(s\right){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }s}}-F_1\left(s\right){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }W}},\\ G_2& =F_2\left(s\right){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }s}}+(1-F_2(s\left)\right){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }W}},\\ G_3& =r{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }r}}+F_3\left(s\right){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }s}}-\left(F_3\right(s)+s+W){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }W}}.\end{array}$ We take $F_1\left(s\right)=F_2\left(s\right)=F_3\left(s\right)=s$ in the symmetries and utilize them to further reduce (14(14) $\theta W_{\mathrm{rr}}+\theta W_s{W}_{\mathrm{rr}}+2\theta W_r{W}_{\mathrm{rs}}+\alpha W_{\mathrm{rrrs}}=0,$(14) ). Thus, from $G_1$, we achieve $W(r,s)=f\left(w\right)-s$, where $w=s{e}^{-r}$. On inserting the current function in Equation (14(14) $\theta W_{\mathrm{rr}}+\theta W_s{W}_{\mathrm{rr}}+2\theta W_r{W}_{\mathrm{rs}}+\alpha W_{\mathrm{rrrs}}=0,$(14) ) purveys the fourth-order NLODE (16) $\begin{array}{rl}& 3\mathrm{\theta w}{f}^{\mathrm{\prime }}(w{)}^2-\alpha f^{\mathrm{\prime }}(w)-7\mathrm{\alpha w}{f}^{\mathrm{\prime \prime }}(w)+3\theta w^2{f}^{\mathrm{\prime }}(w\left)f^{\mathrm{\prime \prime }}\right(w)\\ & -6\alpha w^2{f}^{\mathrm{\prime \prime \prime }}\left(w\right)-\alpha w^3{f}^{\mathrm{\prime \prime \prime \prime }}\left(w\right)=0.\end{array}$(16) Moreover, operator $G_2$ furnishes function $W(r,s)=f\left(w\right)+\ln \left(s\right)-s$, where $w=r$. The obtained function reduces (14(14) $\theta W_{\mathrm{rr}}+\theta W_s{W}_{\mathrm{rr}}+2\theta W_r{W}_{\mathrm{rs}}+\alpha W_{\mathrm{rrrs}}=0,$(14) ) to $f^{\mathrm{\prime \prime }}\left(w\right)=0$, giving solution of NLODE (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) as (17) $u(t,x,y,z)=z+\ln \left(y\right)-y+A_{0}x+A_1,$(17) which is a logarithmic function solution of 3D-genBSMe (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) with integration constants $A_0$ and $A_1$. Next, we engage $G_3$, thus yielding $W(r,s)=1/rf\left(w\right)-s$, where $w=s/r$. On applying the latest expression of $W(r,s)$ in Equation (14(14) $\theta W_{\mathrm{rr}}+\theta W_s{W}_{\mathrm{rr}}+2\theta W_r{W}_{\mathrm{rs}}+\alpha W_{\mathrm{rrrs}}=0,$(14) ) secures the NLODE (18) $\begin{array}{rl}& 8\mathrm{\theta w}{f}^{\mathrm{\prime }}(w{)}^2-24\alpha f^{\mathrm{\prime }}(w)-36\mathrm{\alpha w}{f}^{\mathrm{\prime \prime }}(w)\\ & +3\theta w^2{f}^{\mathrm{\prime }}\left(w\right)f^{\mathrm{\prime \prime }}\left(w\right)-12\alpha w^2{f}^{\mathrm{\prime \prime \prime }}\left(w\right)\\ & +2\mathrm{\theta wf}\left(w\right)f^{\mathrm{\prime \prime }}\left(w\right)+6\mathrm{\theta f}\left(w\right)f^{\mathrm{\prime }}\left(w\right)-\alpha w^3{f}^{\mathrm{\prime \prime \prime \prime }}\left(w\right)=0.\end{array}$(18) Now, we implore operator $Q_2$ whose related characteristic equations solve to provide $U(X,Y,Z)=W(r,s)+Y$, where $r=X$ and $s=Z$, hence transforming (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) to (19) $\theta W_{\mathrm{rr}}+2\theta W_r{W}_{\mathrm{rs}}+\theta W_s{W}_{\mathrm{rr}}+\text{β}{W}_{\mathrm{rrrs}}=0.$(19) We solve the secured NLPDE and then gain a tan-hyperbolic solution of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) as (20) $\begin{array}{rl}& u(t,x,y,z)\\ & =y+{\displaystyle \frac{4\text{β}{B}_2}{\theta }}\tanh (B_{2}x-{\displaystyle \frac{\theta }{4\text{β}{B}_2}}z+B_1)+B_3,\end{array}$(20) with constants $B_1$, $B_2$ alongside $B_3$, arbitrary. Further investigation of (19(19) $\theta W_{\mathrm{rr}}+2\theta W_r{W}_{\mathrm{rs}}+\theta W_s{W}_{\mathrm{rr}}+\text{β}{W}_{\mathrm{rrrs}}=0.$(19) ) gives no new results. We turn attention to $Q_3$ and then, we bestow $U(X,Y,Z)=W(r,s)+X$, where $r=Y$ together with $s=Z$ which gives a trivial solution of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ). Next, in the case of $Q_4$, we secure $U(X,Y,Z)=W(r,s)+\ln \left(Y\right)$, where $r=X$ with $s=Z/Y$. We then engage the function in NLPDE (13(13) $\begin{array}{rl}& \theta U_Y{U}_{\mathrm{XX}}+\theta U_Z{U}_{\mathrm{XX}}+2\theta U_X{U}_{\mathrm{XY}}+2\theta U_X{U}_{\mathrm{XZ}}\\ & +\alpha U_{\mathrm{XXXY}}+\text{β}{U}_{\mathrm{XXXZ}}=0.\end{array}$(13) ) thereby reducing it further to (21) $\begin{array}{rl}& \theta W_{\mathrm{rr}}-2\mathrm{\theta s}{W}_r{W}_{\mathrm{rs}}-\mathrm{\theta s}{W}_s{W}_{\mathrm{rr}}+2\theta W_r{W}_{\mathrm{rs}}\\ & +\theta W_s{W}_{\mathrm{rr}}-\mathrm{\alpha s}{W}_{\mathrm{rrrs}}+\text{β}{W}_{\mathrm{rrrs}}=0.\end{array}$(21) Further examination of Equation (21(21) $\begin{array}{rl}& \theta W_{\mathrm{rr}}-2\mathrm{\theta s}{W}_r{W}_{\mathrm{rs}}-\mathrm{\theta s}{W}_s{W}_{\mathrm{rr}}+2\theta W_r{W}_{\mathrm{rs}}\\ & +\theta W_s{W}_{\mathrm{rr}}-\mathrm{\alpha s}{W}_{\mathrm{rrrs}}+\text{β}{W}_{\mathrm{rrrs}}=0.\end{array}$(21) ) gives the succeeding four operators, viz. $\begin{array}{rl}{G}_1& ={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }r}},G_2={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }W}},\\ G_3& =r{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }r}}+[\ln (s-1)-W]{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }W}},\\ G_4& =(s-1)(\mathrm{\alpha s}-\text{β}){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }s}}+\{(\alpha -\text{β})\ln (s-1)\\ & +\alpha (s-W)+\text{β}W\}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }W}}.\end{array}$ On contemplating the linear combination of $G_1$ and $G_2$ secures a trivial solution of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ). In the case of symmetries $G_3$, we gain $W(r,s)=1/rf\left(w\right)+\ln (s-1)$, where $w=s$. We then insert the function in (21(21) $\begin{array}{rl}& \theta W_{\mathrm{rr}}-2\mathrm{\theta s}{W}_r{W}_{\mathrm{rs}}-\mathrm{\theta s}{W}_s{W}_{\mathrm{rr}}+2\theta W_r{W}_{\mathrm{rs}}\\ & +\theta W_s{W}_{\mathrm{rr}}-\mathrm{\alpha s}{W}_{\mathrm{rrrs}}+\text{β}{W}_{\mathrm{rrrs}}=0.\end{array}$(21) ) and secure the first-order NLODE (22) $\begin{array}{rl}& 6\mathrm{\alpha w}{f}^{\mathrm{\prime }}\left(w\right)-6\text{β}{f}^{\mathrm{\prime }}\left(w\right)+4\mathrm{\theta f}\left(w\right)f^{\mathrm{\prime }}\left(w\right)\\ & -4\mathrm{\theta wf}\left(w\right)f^{\mathrm{\prime }}\left(w\right)=0.\end{array}$(22) The solution to the ordinary differential equation (ODE) leads to dual results (23) $\begin{array}{rl}u(t,x,y,z)& ={\displaystyle \frac{1}{x}}\left\{{\displaystyle \frac{3\mathrm{\alpha z}-3\text{β}y}{2\mathrm{\theta z}-2\mathrm{\theta y}}}\right\}+\ln ({\displaystyle \frac{z}{y}}-1)+\ln \left(y\right),\\ u(t,x,y,z)& =C_0,\end{array}$(23) which satisfy (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) where $C_0$ is an arbitrary constant. We observe that further studies on (13(13) $\begin{array}{rl}& \theta U_Y{U}_{\mathrm{XX}}+\theta U_Z{U}_{\mathrm{XX}}+2\theta U_X{U}_{\mathrm{XY}}+2\theta U_X{U}_{\mathrm{XZ}}\\ & +\alpha U_{\mathrm{XXXY}}+\text{β}{U}_{\mathrm{XXXZ}}=0.\end{array}$(13) ) yield no solution of interest.

2.2.2. Invariant solutions using operator ${\mathcal{P}}_2$

On investigating the vector field ${\mathcal{P}}_2$, it purveys a solution of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) which is of no interest.

2.2.3. Invariant solutions using operator ${\mathcal{P}}_3$

The symmetry ${\mathcal{P}}_2=\mathrm{\partial }/\mathrm{\partial y}$ transforms (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) to the NLPDE of the structure (24) $\theta U_Z{U}_{\mathrm{XX}}+2\theta U_X{U}_{\mathrm{XZ}}-\gamma U_{\mathrm{TX}}+\text{β}{U}_{\mathrm{XXXZ}}=0$(24) through relation $u(t,x,y,z)=U(T,X,Z)$, where $T=t$, $X=x$ and $Z=z$. We solve the gained NLPDE and consequently achieve (25) $\begin{array}{rl}& u(t,x,y,z)\\ & ={\displaystyle \frac{4\text{β}{C}_0}{\theta }}\tanh ({\displaystyle \frac{4\text{β}{C}_0^2{C}_1}{\gamma }}t+C_{0}x+C_{2}z+C_3)+C_4,\end{array}$(25) which is a kink solution of 3D-genBSMe (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) with constants $C_i,i=1,\dots ,4$, arbitrary. On the application of the Lie theoretic method on (24(24) $\theta U_Z{U}_{\mathrm{XX}}+2\theta U_X{U}_{\mathrm{XZ}}-\gamma U_{\mathrm{TX}}+\text{β}{U}_{\mathrm{XXXZ}}=0$(24) ), we secure six operators $\begin{array}{rl}{Q}_1& ={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }T}}+{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }X}}+{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}},Q_2={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }X}}+{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Z}}+{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}},\\ Q_3& ={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }X}}+T{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Z}}+{\displaystyle \frac{1}{2\theta }}(2\theta -\mathrm{\gamma X}){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}},\\ Q_4& =T{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }T}}+(1+{\displaystyle \frac{1}{2}}X){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }X}}+(1-{\displaystyle \frac{1}{2}}U){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}},\\ Q_5& =(1-{\displaystyle \frac{1}{2}}X){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }X}}+Z{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Z}}+(1+{\displaystyle \frac{1}{2}}U){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}},\\ Q_6& ={\displaystyle \frac{1}{2}}{T}^2{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }T}}+(1+{\displaystyle \frac{1}{4}}\mathrm{TX}){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }X}}+{\displaystyle \frac{1}{2}}\mathrm{TZ}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Z}}\\ & +{\displaystyle \frac{1}{4\theta }}(4\theta -\mathrm{\gamma XZ}-\mathrm{\theta TU}){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}}.\end{array}$ On exploring $Q_1$ we obtain group invariant $U(T,X,Z)=W(r,s)+T$, where $r=X-T$ and $s=Z$. The secured invariant solutions transform (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) to the equation (26) $\gamma W_{\mathrm{rr}}+2\theta W_r{W}_{\mathrm{rs}}+\theta W_s{W}_{\mathrm{rr}}+\text{β}{W}_{\mathrm{rrrs}}=0,$(26) which gives no solution of importance. We further explore $Q_2$ and this gives us function $U(T,X,Z)=W(r,s)+X$, where $r=T$ and $s=Z-X$. On inserting the function in (24(24) $\theta U_Z{U}_{\mathrm{XX}}+2\theta U_X{U}_{\mathrm{XZ}}-\gamma U_{\mathrm{TX}}+\text{β}{U}_{\mathrm{XXXZ}}=0$(24) ), we have a reduced form of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) here as (27) $\gamma W_{\mathrm{rs}}-2\theta W_{\mathrm{ss}}+3\theta W_s{W}_{\mathrm{ss}}-\text{β}{W}_{\mathrm{ssss}}=0.$(27) Solving the secured (27(27) $\gamma W_{\mathrm{rs}}-2\theta W_{\mathrm{ss}}+3\theta W_s{W}_{\mathrm{ss}}-\text{β}{W}_{\mathrm{ssss}}=0.$(27) ), one gains a logarithmic-tan-hyperbolic solution of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ); (28) $\begin{array}{rl}& u(t,x,y,z)\\ & =x+{\displaystyle \frac{1}{\theta }}\{\phantom{{\displaystyle \frac{\gamma A_2}{3\theta A_3}}}2\text{β}{A}_3\ln [\tanh (A_{2}t+A_3(z-x\left)+A_1\right)+1]\\ & -{\displaystyle \frac{8\text{β}{A}_3^2}{3\theta }}(z-x)-2\text{β}{A}_3\\ & \times \ln [\tanh (A_{2}t+A_3(z-x)+A_1)-1]\\ & -4\text{β}{A}_3\tanh (A_{2}t\\ & +A_3(z-x)+A_1)-{\displaystyle \frac{\gamma A_2}{3\theta A_3}}(z-x)\}\\ & +{\displaystyle \frac{2}{3}}(z-x)+f_1\left(t\right),\end{array}$(28) where $f_1\left(t\right)$ is a function of t. Using $Q_3$, no solution of interest could be obtained so we give attention to $Q_4$ and in this case, we gain $U(T,X,Z)=1/\sqrt{T}W(r,s)+2$, where we have $r=(X+2)/\sqrt{T}$ and $s=Z$. Using the gained function reduces (24(24) $\theta U_Z{U}_{\mathrm{XX}}+2\theta U_X{U}_{\mathrm{XZ}}-\gamma U_{\mathrm{TX}}+\text{β}{U}_{\mathrm{XXXZ}}=0$(24) ) to (29) $2\gamma W_r+\mathrm{\gamma r}{W}_{\mathrm{rr}}+2\theta W_s{W}_{\mathrm{rr}}+4\theta W_r{W}_{\mathrm{rs}}+2\text{β}{W}_{\mathrm{rrrs}}=0,$(29) which yields four Lie-point symmetries presented here as $\begin{array}{rl}{G}_1& ={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }s}},G_2={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }W}},\\ G_3& =-{\displaystyle \frac{1}{2}}r{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }r}}+s{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }s}}+{\displaystyle \frac{1}{2}}W{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }W}},G_4={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }r}}-{\displaystyle \frac{\gamma }{2\theta }}s{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }W}}.\end{array}$ The linear combination of $G_1$ and $G_2$ purveys $W(r,s)=f\left(w\right)+s$, where $w=r$. On inserting the secured relation in Equation (29(29) $2\gamma W_r+\mathrm{\gamma r}{W}_{\mathrm{rr}}+2\theta W_s{W}_{\mathrm{rr}}+4\theta W_r{W}_{\mathrm{rs}}+2\text{β}{W}_{\mathrm{rrrs}}=0,$(29) ), we achieve the second-order ODE (30) $2\gamma f^{\mathrm{\prime }}\left(w\right)+\mathrm{\gamma w}{f}^{\mathrm{\prime \prime }}\left(w\right)+2\theta f^{\mathrm{\prime \prime }}\left(w\right)=0.$(30) The solution of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) with regard to Equation (30(30) $2\gamma f^{\mathrm{\prime }}\left(w\right)+\mathrm{\gamma w}{f}^{\mathrm{\prime \prime }}\left(w\right)+2\theta f^{\mathrm{\prime \prime }}\left(w\right)=0.$(30) ) furnishes (31) $\begin{array}{rl}& u(t,x,y,z)\\ & ={\displaystyle \frac{1}{\sqrt{t}}}\left\{{\displaystyle \frac{\gamma A_2\sqrt{t}+2\theta A_1\sqrt{t}+\gamma A_{1}x+2\gamma A_1}{2\theta \sqrt{t}+\mathrm{\gamma x}+2\gamma }}+z\right\}+2,\end{array}$(31) where $A_1$ and $A_2$ are integration constants. Now, we explore $G_3$ and that gives us $W(r,s)=1/rf\left(w\right)$, where $w=s{r}^2$. Thus utilizing the relation reduces (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) to (32) $\begin{array}{rl}& 4\mathrm{\theta w}{f}^{\mathrm{\prime }}(w{)}^2+2\gamma f^{\mathrm{\prime }}(w)+4\mathrm{\gamma w}{f}^{\mathrm{\prime \prime }}(w)\\ & +24\mathrm{\theta w}{f}^{\mathrm{\prime }}\left(w\right)f^{\mathrm{\prime \prime }}\left(w\right)+48\text{β}w{f}^{\mathrm{\prime \prime \prime }}\left(w\right)\\ & -8\mathrm{\theta f}\left(w\right)f^{\mathrm{\prime \prime }}\left(w\right)+12\text{β}{f}^{\mathrm{\prime \prime }}\left(w\right)+16\text{β}{w}^2{f}^{\mathrm{\prime \prime \prime \prime }}\left(w\right)=0.\end{array}$(32) Regarding $Q_5$, we secure $U(T,X,Z)=1/(X-2)\left\{W\right(r,s)-2X\}$, where we have $r=T$ and $s=Z(X-2{)}^2$. The obtained function eventually reduces (24(24) $\theta U_Z{U}_{\mathrm{XX}}+2\theta U_X{U}_{\mathrm{XZ}}-\gamma U_{\mathrm{TX}}+\text{β}{U}_{\mathrm{XXXZ}}=0$(24) ) to (33) $\begin{array}{rl}& \gamma W_r-2\mathrm{\gamma s}{W}_{\mathrm{rs}}+16\mathrm{\theta s}{W}_{\mathrm{ss}}+6\text{β}s{W}_{\mathrm{ss}}-4\mathrm{\theta sW}{W}_{\mathrm{ss}}\\ & +2\mathrm{\theta s}{W}_s^2+24\text{β}{s}^2{W}_{\mathrm{sss}}+12\theta s^2{W}_s{W}_{\mathrm{ss}}\\ & +8\text{β}{s}^3{W}_{\mathrm{ssss}}=0.\end{array}$(33) Now, we examine $Q_6$, and this gives the invariant (34) $\begin{array}{rl}U& =\frac{1}{\sqrt{T}}W(r,s)-\frac{1}{T}(\frac{\gamma }{2\theta }\mathrm{XZ}+4)+\frac{2\gamma }{\theta T^2}Z,\\ r& =\frac{1}{3\sqrt[3]{T^2}}(3\mathrm{TX}+4),s=\frac{Z}{T}.\end{array}$(34) Inserting the expression of U from (34(34) $\begin{array}{rl}U& =\frac{1}{\sqrt{T}}W(r,s)-\frac{1}{T}(\frac{\gamma }{2\theta }\mathrm{XZ}+4)+\frac{2\gamma }{\theta T^2}Z,\\ r& =\frac{1}{3\sqrt[3]{T^2}}(3\mathrm{TX}+4),s=\frac{Z}{T}.\end{array}$(34) ) in (24(24) $\theta U_Z{U}_{\mathrm{XX}}+2\theta U_X{U}_{\mathrm{XZ}}-\gamma U_{\mathrm{TX}}+\text{β}{U}_{\mathrm{XXXZ}}=0$(24) ) furnishes the NLPDE (35) $2\theta W_r{W}_{\mathrm{rs}}+\theta W_s{W}_{\mathrm{rr}}+\text{β}{W}_{\mathrm{rrrs}}=0.$(35) Solving the achieved NLPDE provides a solution of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) as (36) $\begin{array}{rl}u(t,x,y,z)& =\frac{1}{\sqrt{t}}{f}_1\left(\frac{1}{3\sqrt[3]{t^2}}(3\mathrm{tx}+4)\right)f_2\left(\frac{z}{t}\right)\\ & -\frac{1}{t}(\frac{\gamma }{2\theta }\mathrm{xz}+4)+\frac{2\gamma }{\theta t^2}z,\end{array}$(36) where functions $f_1$ and $f_2$ are arbitrary and are depending on their arguments.

2.2.4. Invariant solutions using operator ${\mathcal{P}}_4$

Investigating Lie operator ${\mathcal{P}}_4=\mathrm{\partial }/\mathrm{\partial z}$, we gain solution to its Lagrangian system as $u(t,x,y,z)=U(T,X,Y)$, where $T=t$, $X=x$ and $Y=y$. Thus transforming (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) to (37) $\theta U_Y{U}_{\mathrm{XX}}+2\theta U_X{U}_{\mathrm{XY}}-\gamma U_{\mathrm{TX}}+\alpha U_{\mathrm{XXXY}}=0.$(37) On the application of the Lie theoretic method on (24(24) $\theta U_Z{U}_{\mathrm{XX}}+2\theta U_X{U}_{\mathrm{XZ}}-\gamma U_{\mathrm{TX}}+\text{β}{U}_{\mathrm{XXXZ}}=0$(24) ), we secure six operators $\begin{array}{rl}{Q}_1& ={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }T}}+{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }X}}+{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}},Q_2={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }X}}+{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Y}}+{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}},\\ Q_3& ={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }X}}+T{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Y}}+{\displaystyle \frac{1}{2\theta }}(2\theta -\mathrm{\gamma X}){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}},\\ Q_4& =T{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }T}}+(1+{\displaystyle \frac{1}{2}}X){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }X}}+(1-{\displaystyle \frac{1}{2}}U){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}},\\ Q_5& =(1-{\displaystyle \frac{1}{2}}X){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }X}}+Y{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Y}}+(1+{\displaystyle \frac{1}{2}}U){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}},\\ Q_6& ={\displaystyle \frac{1}{2}}{T}^2{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }T}}+(1+{\displaystyle \frac{1}{4}}\mathrm{TX}){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }X}}+{\displaystyle \frac{1}{2}}\mathrm{TY}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Y}}\\ & +{\displaystyle \frac{1}{4\theta }}(4\theta -\mathrm{\gamma XY}-\mathrm{\theta TU}){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}}.\end{array}$ On exploring $Q_1$ we obtain group invariant $U(T,X,Y)=W(r,s)+T$, where $r=X-T$ and $s=Y$. The secured invariant solutions transform (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) to the equation (38) $\gamma W_{\mathrm{rr}}+2\theta W_r{W}_{\mathrm{rs}}+\theta W_s{W}_{\mathrm{rr}}+\text{β}{W}_{\mathrm{rrrs}}=0,$(38) which consequently solves to give a solution of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) particularly here as (39) $\begin{array}{rl}& u(t,x,y,z)\\ & =t+{\displaystyle \frac{4\alpha C_2}{\theta }}\tanh (C_1-C_2(t-x)-{\displaystyle \frac{\gamma }{4\alpha C_2}}y)+C_3,\end{array}$(39) which is another topological kink soliton solution of 3D-genBSMe (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) with arbitrary constants $C_i,i=1,\dots ,3$. We further explore $Q_2$ and this fetches us relation $U(T,X,Y)=W(r,s)+X$, where $r=T$ as well as $s=Y-X$. On inserting the supplied function in (37(37) $\theta U_Y{U}_{\mathrm{XX}}+2\theta U_X{U}_{\mathrm{XY}}-\gamma U_{\mathrm{TX}}+\alpha U_{\mathrm{XXXY}}=0.$(37) ), we have a transformed structure of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) in this regard as (40) $\gamma W_{\mathrm{rs}}-2\theta W_{\mathrm{ss}}+3\theta W_s{W}_{\mathrm{ss}}-\alpha W_{\mathrm{ssss}}=0.$(40) Solving the achieved (40(40) $\gamma W_{\mathrm{rs}}-2\theta W_{\mathrm{ss}}+3\theta W_s{W}_{\mathrm{ss}}-\alpha W_{\mathrm{ssss}}=0.$(40) ), one gains a solution of 3D-genBSMe (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) here as (41) $\begin{array}{rl}& u(t,x,y,z)\\ & =x+{\displaystyle \frac{1}{\theta }}\{2\alpha A_3\\ & \times \ln \left\{{\displaystyle \frac{[\tanh (A_{2}t+A_3(y-x)+A_1)+1]}{[\tanh (A_{2}t+A_3(y-x)+A_1)-1]}}\right\}\\ & -{\displaystyle \frac{8\alpha A_3^2}{3\theta }}(y-x)\\ & -4\alpha A_3\tanh (A_{2}t+A_3(y-x)+A_1)\\ & -{\displaystyle \frac{\gamma A_2}{3\theta A_3}}(y-x)\}+{\displaystyle \frac{2}{3}}(y-x)+f_1\left(t\right),\end{array}$(41) which is a logarithmic-tan-hyperbolic function solution with arbitrary function $f_1\left(t\right)$ where $f_1$ depends on t.

On utilizing $Q_3$, no solution of interest could be obtained so we give attention to $Q_4$ and in this case, we gain $U(T,X,Y)=1/\sqrt{T}W(r,s)+2$, where one has $r=(X+2)/\sqrt{T}$ and $s=Y$. Using the achieved relation reduces (37(37) $\theta U_Y{U}_{\mathrm{XX}}+2\theta U_X{U}_{\mathrm{XY}}-\gamma U_{\mathrm{TX}}+\alpha U_{\mathrm{XXXY}}=0.$(37) ) to (42) $2\gamma W_r+\mathrm{\gamma r}{W}_{\mathrm{rr}}+2\theta W_s{W}_{\mathrm{rr}}+4\theta W_r{W}_{\mathrm{rs}}+2\alpha W_{\mathrm{rrrs}}=0,$(42) which yields four Lie-point symmetries explicated here as $\begin{array}{rl}{G}_1& ={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }s}},G_2={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }W}},\\ G_3& =-{\displaystyle \frac{1}{2}}r{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }r}}+s{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }s}}+{\displaystyle \frac{1}{2}}W{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }W}},G_4={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }r}}-{\displaystyle \frac{\gamma }{2\theta }}s{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }W}}.\end{array}$ The linear combination of $G_1$ and $G_2$ purveys $W(r,s)=f\left(w\right)+s$, where $w=r$. On invoking the obtained relation in (42(42) $2\gamma W_r+\mathrm{\gamma r}{W}_{\mathrm{rr}}+2\theta W_s{W}_{\mathrm{rr}}+4\theta W_r{W}_{\mathrm{rs}}+2\alpha W_{\mathrm{rrrs}}=0,$(42) ), we achieve an ODE of order 2, that is (43) $2\gamma f^{\mathrm{\prime }}\left(w\right)+\mathrm{\gamma w}{f}^{\mathrm{\prime \prime }}\left(w\right)+2\theta f^{\mathrm{\prime \prime }}\left(w\right)=0.$(43) Consequently, the solution of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) with regard to Equation (30(30) $2\gamma f^{\mathrm{\prime }}\left(w\right)+\mathrm{\gamma w}{f}^{\mathrm{\prime \prime }}\left(w\right)+2\theta f^{\mathrm{\prime \prime }}\left(w\right)=0.$(30) ) purveys (44) $\begin{array}{rl}& u(t,x,y,z)\\ & ={\displaystyle \frac{1}{\sqrt{t}}}\left\{{\displaystyle \frac{\gamma B_2\sqrt{t}+2\theta B_1\sqrt{t}+\gamma B_{1}x+2\gamma B_1}{2\theta \sqrt{t}+\mathrm{\gamma x}+2\gamma }}+y\right\}+2,\end{array}$(44) with $B_1$ and $B_2$ as constants of integration. Now, we explore $G_3$ and that gives us $W(r,s)=1/rf\left(w\right)$, where $w=s{r}^2$. Thus on engaging the relation reduces (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) to (45) $\begin{array}{rl}& 4\theta f^{\mathrm{\prime }}(w{)}^2+2\gamma f^{\mathrm{\prime }}(w)+4\mathrm{\gamma w}{f}^{\mathrm{\prime \prime }}(w)\\ & +24\mathrm{\theta w}{f}^{\mathrm{\prime }}\left(w\right)f^{\mathrm{\prime \prime }}\left(w\right)+48\mathrm{\alpha w}{f}^{\mathrm{\prime \prime \prime }}\left(w\right)\\ & -8\mathrm{\theta f}\left(w\right)f^{\mathrm{\prime \prime }}\left(w\right)+12\alpha f^{\mathrm{\prime \prime }}\left(w\right)+16\alpha w^2{f}^{\mathrm{\prime \prime \prime \prime }}\left(w\right)=0.\end{array}$(45) No solution(s) of interest could be achieved upon further examination of Lie point symmetry ${\mathcal{P}}_4$.

2.2.5. Invariant solutions using operators ${\mathcal{P}}_5$ and ${\mathcal{P}}_6$

The vector fields ${\mathcal{P}}_5$ and ${\mathcal{P}}_6$ do not possess invariant solutions.

2.2.6. Invariant solutions using operator ${\mathcal{P}}_7$

Symmetry reduction via ${\mathcal{P}}_7=u\mathrm{\partial }/\mathrm{\partial u}-2t\mathrm{\partial }/\mathrm{\partial t}-x\mathrm{\partial }/\mathrm{\partial x}$ can be achieved if (46) $\begin{array}{rl}& u(t,x,y,z)\\ & ={\displaystyle \frac{1}{\sqrt{t}}}U(X,Y,Z),\text{where we have}X={\displaystyle \frac{x}{\sqrt{t}}},\\ & Y=y,\mathrm{alongside}Z=z,\end{array}$(46) which is the solution to the corresponding characteristic system of ${\mathcal{P}}_7$. Therefore, using the function explicated in (46(46) $\begin{array}{rl}& u(t,x,y,z)\\ & ={\displaystyle \frac{1}{\sqrt{t}}}U(X,Y,Z),\text{where we have}X={\displaystyle \frac{x}{\sqrt{t}}},\\ & Y=y,\mathrm{alongside}Z=z,\end{array}$(46) ) transforms 3D-genBSMe (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) to NLPDE (47) $\begin{array}{rl}& 2\gamma U_X+\mathrm{\gamma X}{U}_{\mathrm{XX}}+2\theta U_Y{U}_{\mathrm{XX}}+2\theta U_Z{U}_{\mathrm{XX}}\\ & +4\theta U_X{U}_{\mathrm{XY}}+4\theta U_X{U}_{\mathrm{XZ}}\\ & +2\alpha U_{\mathrm{XXXY}}+2\text{β}{U}_{\mathrm{XXXZ}}=0.\end{array}$(47) Lie-symmetries analysis of Equation (47(47) $\begin{array}{rl}& 2\gamma U_X+\mathrm{\gamma X}{U}_{\mathrm{XX}}+2\theta U_Y{U}_{\mathrm{XX}}+2\theta U_Z{U}_{\mathrm{XX}}\\ & +4\theta U_X{U}_{\mathrm{XY}}+4\theta U_X{U}_{\mathrm{XZ}}\\ & +2\alpha U_{\mathrm{XXXY}}+2\text{β}{U}_{\mathrm{XXXZ}}=0.\end{array}$(47) ) yields six vector fields of the structure $\begin{array}{rl}{Q}_1& ={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Z}}+{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}},Q_2={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Y}}+{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}},\\ Q_3& ={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }T}}+(1-{\displaystyle \frac{\gamma }{2\theta }}Y){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}},\\ Q_4& =-{\displaystyle \frac{1}{2}}T{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }T}}+X{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }X}}+Z{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Z}}+(1+{\displaystyle \frac{1}{2}}U){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}},\\ Q_5& =-{\displaystyle \frac{\alpha }{2\text{β}}}T{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }T}}+{\displaystyle \frac{1}{\text{β}}}[\text{β}Y+\alpha (Y-Z)]{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Y}}\\ & +Y{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Z}}+(1+U-{\displaystyle \frac{\alpha }{2\text{β}}}U){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}}.\end{array}$ Group-invariant solution of $Q_1$ yields $U(X,Y,Z)=W(r,s)+Z$, where we have $r=X$ and $s=Y$. On using, the calculated function further reduces (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) to (48) $\begin{array}{rl}& \mathrm{\gamma r}{W}_{\mathrm{rr}}+2\gamma W_r+2\theta W_{\mathrm{rr}}+4\theta W_r{W}_{\mathrm{rs}}\\ & +2\theta W_s{W}_{\mathrm{rr}}+2\alpha W_{\mathrm{rrrs}}=0.\end{array}$(48) Thus producing four Lie operators $\begin{array}{rl}{G}_1& ={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }s}},G_2={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }W}},G_3={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }r}}-{\displaystyle \frac{\gamma }{2\theta }}s{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }W}},\\ G_4& =-{\displaystyle \frac{1}{2}}r{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }r}}+s{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }s}}+({\displaystyle \frac{1}{2}}W-{\displaystyle \frac{1}{2}}s){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }W}}.\end{array}$ On solving $G=G_1+G_2$, we have $W(r,s)=f\left(w\right)+s$, where $w=r$. The function further reduces Equation (48(48) $\begin{array}{rl}& \mathrm{\gamma r}{W}_{\mathrm{rr}}+2\gamma W_r+2\theta W_{\mathrm{rr}}+4\theta W_r{W}_{\mathrm{rs}}\\ & +2\theta W_s{W}_{\mathrm{rr}}+2\alpha W_{\mathrm{rrrs}}=0.\end{array}$(48) ) to (49) $2\gamma f^{\mathrm{\prime }}\left(w\right)+\mathrm{\gamma w}{f}^{\mathrm{\prime \prime }}\left(w\right)+4\theta f^{\mathrm{\prime \prime }}\left(w\right)=0.$(49) Eventually, the solution of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) with regard to ODE (48(48) $\begin{array}{rl}& \mathrm{\gamma r}{W}_{\mathrm{rr}}+2\gamma W_r+2\theta W_{\mathrm{rr}}+4\theta W_r{W}_{\mathrm{rs}}\\ & +2\theta W_s{W}_{\mathrm{rr}}+2\alpha W_{\mathrm{rrrs}}=0.\end{array}$(48) ) gives (50) $u(t,x,y,z)={\displaystyle \frac{1}{\sqrt{t}}}\{{\displaystyle \frac{\gamma C_2\sqrt{t}+4\theta C_1\sqrt{t}+\gamma C_{1}x}{4\theta \sqrt{t}+\mathrm{\gamma x}}}+y\}$(50) with integration constants $C_1$ alongside $C_2$. In addition, $G_3$ gives a trivial solution of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) and $G_4$ supplies the invariant solution group $W(r,s)=1/rf\left(w\right)-s$, where $w=s{r}^2$. On inserting the relation in Equation (48(48) $\begin{array}{rl}& \mathrm{\gamma r}{W}_{\mathrm{rr}}+2\gamma W_r+2\theta W_{\mathrm{rr}}+4\theta W_r{W}_{\mathrm{rs}}\\ & +2\theta W_s{W}_{\mathrm{rr}}+2\alpha W_{\mathrm{rrrs}}=0.\end{array}$(48) ), one gets the fourth-order NLODE (51) $\begin{array}{rl}& 4\theta f^{\mathrm{\prime }}(w{)}^2+2\gamma f^{\mathrm{\prime }}(w)+4\mathrm{\gamma w}{f}^{\mathrm{\prime \prime }}(w)\\ & +24\mathrm{\theta w}{f}^{\mathrm{\prime }}\left(w\right)f^{\mathrm{\prime \prime }}\left(w\right)+48\mathrm{\alpha w}{f}^{\mathrm{\prime \prime \prime }}\left(w\right)\\ & -8\mathrm{\theta f}\left(w\right)f^{\mathrm{\prime \prime }}\left(w\right)+12\alpha f^{\mathrm{\prime \prime }}\left(w\right)+16\alpha w^2{f}^{\mathrm{\prime \prime \prime \prime }}\left(w\right)=0.\end{array}$(51) Further investigation of symmetry ${\mathcal{P}}_7$ produces no other solution(s) of importance.

2.2.7. Invariant solutions using operator ${\mathcal{P}}_8$

The Lie-point symmetry ${\mathcal{P}}_8=t\mathrm{\partial }/\mathrm{\partial t}+y\mathrm{\partial }/\mathrm{\partial y}+z\mathrm{\partial }/\mathrm{\partial z}$ furnishes invariant (52) $\begin{array}{rl}u(t,x,y,z)& =U(X,Y,Z),\text{where we have}X=x,\\ & Y={\displaystyle \frac{y}{t}},\mathrm{alongside}Z={\displaystyle \frac{z}{t}}\end{array}$(52) that ultimately transmutes 3D-genBSMe (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) to the NLPDE of the format (53) $\begin{array}{rl}& \mathrm{\gamma Y}{U}_{\mathrm{XY}}+\mathrm{\gamma Z}{U}_{\mathrm{XZ}}+\theta U_Y{U}_{\mathrm{XX}}+\theta U_Z{U}_{\mathrm{XX}}+2\theta U_X{U}_{\mathrm{XY}}\\ & +2\theta U_X{U}_{\mathrm{XZ}}+\alpha U_{\mathrm{XXXY}}+\text{β}{U}_{\mathrm{XXXZ}}=0.\end{array}$(53) Lie symmetry analysis of Equation (53(53) $\begin{array}{rl}& \mathrm{\gamma Y}{U}_{\mathrm{XY}}+\mathrm{\gamma Z}{U}_{\mathrm{XZ}}+\theta U_Y{U}_{\mathrm{XX}}+\theta U_Z{U}_{\mathrm{XX}}+2\theta U_X{U}_{\mathrm{XY}}\\ & +2\theta U_X{U}_{\mathrm{XZ}}+\alpha U_{\mathrm{XXXY}}+\text{β}{U}_{\mathrm{XXXZ}}=0.\end{array}$(53) ) yields six vector fields of the structure $\begin{array}{rl}{Q}_1& =-{\displaystyle \frac{1}{2}}X{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }X}}+Y{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Y}}+Z{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Z}}+(1+{\displaystyle \frac{1}{2}}U){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}},\\ Q_2& ={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }X}}+{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}},Q_3={\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Y}}+{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Z}}+(1-{\displaystyle \frac{\gamma }{2\theta }}X){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}},\\ Q_4& =-{\displaystyle \frac{\alpha }{2\text{β}}}X{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }X}}+{\displaystyle \frac{1}{\text{β}}}[\text{β}Y+\alpha (Y-Z)]{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Y}}\\ & +Y{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }Z}}+(1+U-{\displaystyle \frac{\alpha }{2\text{β}}}U){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }U}}.\end{array}$ We explore $Q_1$ and this occasions the relation $U(X,Y,Z)=1/XW(r,s)-2$, where one has $r=X^{2}Y$ together with $s=X^{2}Z$. On engaging, the gained function further transforms (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) to (54) $\begin{array}{rl}& 24\text{β}{r}^{2}s{W}_{\mathrm{rrss}}+24\text{β}r{s}^2{W}_{\mathrm{sssr}}+4\mathrm{\gamma rs}{W}_{\mathrm{rs}}+2\mathrm{\theta s}{W}_r{W}_s\\ & -4\mathrm{\theta sW}{W}_{\mathrm{rs}}-4\mathrm{\theta sW}{W}_{\mathrm{ss}}\\ & +48\mathrm{\alpha rs}{W}_{\mathrm{rrs}}+48\text{β}\mathrm{rs}{W}_{\mathrm{rss}}+4\theta s^2{W}_r{W}_{\mathrm{ss}}\\ & +8\theta s^2{W}_s{W}_{\mathrm{rs}}+12\theta s^2{W}_s{W}_{\mathrm{ss}}\\ & +24\alpha r^{2}s{W}_{\mathrm{rrrs}}+24\mathrm{\alpha r}{s}^2{W}_{\mathrm{rrss}}+8\text{β}{s}^3{W}_{\mathrm{ssss}}\\ & +2\gamma s^2{W}_{\mathrm{ss}}+2\mathrm{\theta s}{W}_s^2+24\alpha s^2{W}_{\mathrm{rss}}\\ & +24\text{β}{s}^2{W}_{\mathrm{sss}}+\mathrm{\gamma s}{W}_s+6\mathrm{\alpha s}{W}_{\mathrm{rs}}+6\text{β}s{W}_{\mathrm{ss}}\\ & +8\alpha r^3{W}_{\mathrm{rrrr}}+2\gamma r^2{W}_{\mathrm{rr}}+8\text{β}{r}^3{W}_{\mathrm{rrrs}}\\ & +24\alpha r^2{W}_{\mathrm{rrr}}+\mathrm{\gamma r}{W}_r+6\text{β}r{W}_{\mathrm{rs}}+6\mathrm{\alpha r}{W}_{\mathrm{rr}}\\ & +24\text{β}{r}^2{W}_{\mathrm{rrs}}+2\mathrm{\theta r}{W}_r^2+16\mathrm{\theta rs}{W}_r{W}_{\mathrm{rs}}\\ & +8\mathrm{\theta rs}{W}_r{W}_{\mathrm{ss}}+16\mathrm{\theta rs}{W}_s{W}_{\mathrm{rs}}+8\mathrm{\theta rs}{W}_s{W}_{\mathrm{rr}}\\ & +8\alpha s^3{W}_{\mathrm{rsss}}-4\mathrm{\theta rW}{W}_{\mathrm{rr}}\\ & +8\theta r^2{W}_r{W}_{\mathrm{rs}}+12\theta r^2{W}_r{W}_{\mathrm{rr}}+4\theta r^2{W}_s{W}_{\mathrm{rr}}\\ & +2\mathrm{\theta r}{W}_r{W}_s-4\mathrm{\theta rW}{W}_{\mathrm{rs}}=0.\end{array}$(54) Further study reveals that no solution of importance could be achieved from $Q_2,\dots ,Q_4$.

2.2.8. Invariant solutions using operator ${\mathcal{P}}_9$

Contemplating symmetry ${\mathcal{P}}_9=-2\mathrm{\theta t}\mathrm{\partial }/\mathrm{\partial y}-2\mathrm{\theta t}\mathrm{\partial }/\mathrm{\partial z}+\mathrm{\gamma x}\mathrm{\partial }/\mathrm{\partial u}$, one has relation (55) $\begin{array}{rl}u(t,x,y,z)& =U(T,X,Y)-{\displaystyle \frac{\mathrm{\gamma xy}}{2\mathrm{\theta t}}},\mathrm{having}T=t,\\ & X=x,\mathrm{aswellas}Y=z-y.\end{array}$(55) In utilizing the gained function in (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) reduces it to an equation of the form (56) $\begin{array}{rl}& 2\text{β}T{U}_{\mathrm{XXXY}}-2\mathrm{\alpha T}{U}_{\mathrm{XXXY}}-2\mathrm{\gamma T}{U}_{\mathrm{XY}}\\ & -\mathrm{\gamma X}{U}_{\mathrm{XX}}-2\gamma U_X=0.\end{array}$(56)

Remark 2.1

We discovered that no viable results could be achieved through operators ${\mathcal{P}}_{10}$ and ${\mathcal{P}}_{11}$ with $G\left(t\right)=t$.

2.2.9. Invariant solutions using linear combination of ${\mathcal{P}}_1,{\mathcal{P}}_2,\dots ,{\mathcal{P}}_4$

We do the linear combination of differential generators ${\mathcal{P}}_1$, ${\mathcal{P}}_2$, ${\mathcal{P}}_3$ and ${\mathcal{P}}_4$ as $\mathcal{P}=ϵ{\mathcal{P}}_1+{\mathcal{P}}_2+{\mathcal{P}}_3+{\mathcal{P}}_4$, constant $ϵ\ne 0$, to reduce the 3D-genBSMe (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) to an NLPDE in three independent variables. Hence solution to the related characteristic equations to $\mathcal{P}$ produces the invariants: (57) $f=t-\mathrm{ϵx},g=t-\mathrm{ϵy},h=x-z,\mathrm{\Psi }=u.$(57) In reference to Ψ as new dependent variables as well as taking f, g as well as h as new independent variables, 3D-genBSMe (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) then transmutes into (58) $\begin{array}{rl}& \alpha {ϵ}^4{\mathrm{\Psi }}_{\mathrm{fffg}}-2\theta {ϵ}^3{\mathrm{\Psi }}_f{\mathrm{\Psi }}_{\mathrm{fg}}-\theta {ϵ}^3{\mathrm{\Psi }}_g{\mathrm{\Psi }}_{\mathrm{ff}}-3\alpha {ϵ}^3{\mathrm{\Psi }}_{\mathrm{ffgh}}\\ & +\text{β}{ϵ}^3{\mathrm{\Psi }}_{\mathrm{fffh}}+2\theta {ϵ}^2{\mathrm{\Psi }}_{\mathrm{gh}}{\mathrm{\Psi }}_f\\ & +2\theta {ϵ}^2{\mathrm{\Psi }}_g{\mathrm{\Psi }}_{\mathrm{fh}}-2\theta {ϵ}^2{\mathrm{\Psi }}_f{\mathrm{\Psi }}_{\mathrm{fh}}+2\theta {ϵ}^2{\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{fg}}\\ & +3\alpha {ϵ}^2{\mathrm{\Psi }}_{\mathrm{fghh}}-\theta {ϵ}^2{\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{ff}}-3\text{β}{ϵ}^2{\mathrm{\Psi }}_{\mathrm{ffhh}}\\ & -\mathrm{\theta ϵ}{\mathrm{\Psi }}_{\mathrm{hh}}{\mathrm{\Psi }}_g-2\mathrm{\theta ϵ}{\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{gh}}-\mathrm{\alpha ϵ}{\mathrm{\Psi }}_{\mathrm{ghhh}}\\ & +2\mathrm{\theta ϵ}{\mathrm{\Psi }}_{\mathrm{hh}}{\mathrm{\Psi }}_f+4\mathrm{\theta ϵ}{\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{fh}}+3\text{β}ϵ{\mathrm{\Psi }}_{\mathrm{fhhh}}+\mathrm{\gamma ϵ}{\mathrm{\Psi }}_{\mathrm{fg}}\\ & +\mathrm{\gamma ϵ}{\mathrm{\Psi }}_{\mathrm{ff}}-3\theta {\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{hh}}-\text{β}{\mathrm{\Psi }}_{\mathrm{hhhh}}\\ & -\gamma {\mathrm{\Psi }}_{\mathrm{gh}}-\gamma {\mathrm{\Psi }}_{\mathrm{fh}}=0.\end{array}$(58) Clearly, NLPDEs (58(58) $\begin{array}{rl}& \alpha {ϵ}^4{\mathrm{\Psi }}_{\mathrm{fffg}}-2\theta {ϵ}^3{\mathrm{\Psi }}_f{\mathrm{\Psi }}_{\mathrm{fg}}-\theta {ϵ}^3{\mathrm{\Psi }}_g{\mathrm{\Psi }}_{\mathrm{ff}}-3\alpha {ϵ}^3{\mathrm{\Psi }}_{\mathrm{ffgh}}\\ & +\text{β}{ϵ}^3{\mathrm{\Psi }}_{\mathrm{fffh}}+2\theta {ϵ}^2{\mathrm{\Psi }}_{\mathrm{gh}}{\mathrm{\Psi }}_f\\ & +2\theta {ϵ}^2{\mathrm{\Psi }}_g{\mathrm{\Psi }}_{\mathrm{fh}}-2\theta {ϵ}^2{\mathrm{\Psi }}_f{\mathrm{\Psi }}_{\mathrm{fh}}+2\theta {ϵ}^2{\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{fg}}\\ & +3\alpha {ϵ}^2{\mathrm{\Psi }}_{\mathrm{fghh}}-\theta {ϵ}^2{\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{ff}}-3\text{β}{ϵ}^2{\mathrm{\Psi }}_{\mathrm{ffhh}}\\ & -\mathrm{\theta ϵ}{\mathrm{\Psi }}_{\mathrm{hh}}{\mathrm{\Psi }}_g-2\mathrm{\theta ϵ}{\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{gh}}-\mathrm{\alpha ϵ}{\mathrm{\Psi }}_{\mathrm{ghhh}}\\ & +2\mathrm{\theta ϵ}{\mathrm{\Psi }}_{\mathrm{hh}}{\mathrm{\Psi }}_f+4\mathrm{\theta ϵ}{\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{fh}}+3\text{β}ϵ{\mathrm{\Psi }}_{\mathrm{fhhh}}+\mathrm{\gamma ϵ}{\mathrm{\Psi }}_{\mathrm{fg}}\\ & +\mathrm{\gamma ϵ}{\mathrm{\Psi }}_{\mathrm{ff}}-3\theta {\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{hh}}-\text{β}{\mathrm{\Psi }}_{\mathrm{hhhh}}\\ & -\gamma {\mathrm{\Psi }}_{\mathrm{gh}}-\gamma {\mathrm{\Psi }}_{\mathrm{fh}}=0.\end{array}$(58) ) exists in terms of three independent variables. Engaging the symmetries of (58(58) $\begin{array}{rl}& \alpha {ϵ}^4{\mathrm{\Psi }}_{\mathrm{fffg}}-2\theta {ϵ}^3{\mathrm{\Psi }}_f{\mathrm{\Psi }}_{\mathrm{fg}}-\theta {ϵ}^3{\mathrm{\Psi }}_g{\mathrm{\Psi }}_{\mathrm{ff}}-3\alpha {ϵ}^3{\mathrm{\Psi }}_{\mathrm{ffgh}}\\ & +\text{β}{ϵ}^3{\mathrm{\Psi }}_{\mathrm{fffh}}+2\theta {ϵ}^2{\mathrm{\Psi }}_{\mathrm{gh}}{\mathrm{\Psi }}_f\\ & +2\theta {ϵ}^2{\mathrm{\Psi }}_g{\mathrm{\Psi }}_{\mathrm{fh}}-2\theta {ϵ}^2{\mathrm{\Psi }}_f{\mathrm{\Psi }}_{\mathrm{fh}}+2\theta {ϵ}^2{\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{fg}}\\ & +3\alpha {ϵ}^2{\mathrm{\Psi }}_{\mathrm{fghh}}-\theta {ϵ}^2{\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{ff}}-3\text{β}{ϵ}^2{\mathrm{\Psi }}_{\mathrm{ffhh}}\\ & -\mathrm{\theta ϵ}{\mathrm{\Psi }}_{\mathrm{hh}}{\mathrm{\Psi }}_g-2\mathrm{\theta ϵ}{\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{gh}}-\mathrm{\alpha ϵ}{\mathrm{\Psi }}_{\mathrm{ghhh}}\\ & +2\mathrm{\theta ϵ}{\mathrm{\Psi }}_{\mathrm{hh}}{\mathrm{\Psi }}_f+4\mathrm{\theta ϵ}{\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{fh}}+3\text{β}ϵ{\mathrm{\Psi }}_{\mathrm{fhhh}}+\mathrm{\gamma ϵ}{\mathrm{\Psi }}_{\mathrm{fg}}\\ & +\mathrm{\gamma ϵ}{\mathrm{\Psi }}_{\mathrm{ff}}-3\theta {\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{hh}}-\text{β}{\mathrm{\Psi }}_{\mathrm{hhhh}}\\ & -\gamma {\mathrm{\Psi }}_{\mathrm{gh}}-\gamma {\mathrm{\Psi }}_{\mathrm{fh}}=0.\end{array}$(58) ), it's altered to another NLPDE with regard to two independent variables. Equation (58(58) $\begin{array}{rl}& \alpha {ϵ}^4{\mathrm{\Psi }}_{\mathrm{fffg}}-2\theta {ϵ}^3{\mathrm{\Psi }}_f{\mathrm{\Psi }}_{\mathrm{fg}}-\theta {ϵ}^3{\mathrm{\Psi }}_g{\mathrm{\Psi }}_{\mathrm{ff}}-3\alpha {ϵ}^3{\mathrm{\Psi }}_{\mathrm{ffgh}}\\ & +\text{β}{ϵ}^3{\mathrm{\Psi }}_{\mathrm{fffh}}+2\theta {ϵ}^2{\mathrm{\Psi }}_{\mathrm{gh}}{\mathrm{\Psi }}_f\\ & +2\theta {ϵ}^2{\mathrm{\Psi }}_g{\mathrm{\Psi }}_{\mathrm{fh}}-2\theta {ϵ}^2{\mathrm{\Psi }}_f{\mathrm{\Psi }}_{\mathrm{fh}}+2\theta {ϵ}^2{\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{fg}}\\ & +3\alpha {ϵ}^2{\mathrm{\Psi }}_{\mathrm{fghh}}-\theta {ϵ}^2{\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{ff}}-3\text{β}{ϵ}^2{\mathrm{\Psi }}_{\mathrm{ffhh}}\\ & -\mathrm{\theta ϵ}{\mathrm{\Psi }}_{\mathrm{hh}}{\mathrm{\Psi }}_g-2\mathrm{\theta ϵ}{\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{gh}}-\mathrm{\alpha ϵ}{\mathrm{\Psi }}_{\mathrm{ghhh}}\\ & +2\mathrm{\theta ϵ}{\mathrm{\Psi }}_{\mathrm{hh}}{\mathrm{\Psi }}_f+4\mathrm{\theta ϵ}{\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{fh}}+3\text{β}ϵ{\mathrm{\Psi }}_{\mathrm{fhhh}}+\mathrm{\gamma ϵ}{\mathrm{\Psi }}_{\mathrm{fg}}\\ & +\mathrm{\gamma ϵ}{\mathrm{\Psi }}_{\mathrm{ff}}-3\theta {\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{hh}}-\text{β}{\mathrm{\Psi }}_{\mathrm{hhhh}}\\ & -\gamma {\mathrm{\Psi }}_{\mathrm{gh}}-\gamma {\mathrm{\Psi }}_{\mathrm{fh}}=0.\end{array}$(58) ) produces the eight generators which are $\begin{array}{rl}{\mathrm{\Gamma }}_1& =\frac{\mathrm{\partial }}{\mathrm{\partial }f},{\mathrm{\Gamma }}_2=\frac{\mathrm{\partial }}{\mathrm{\partial }g},{\mathrm{\Gamma }}_3=\frac{\mathrm{\partial }}{\mathrm{\partial }h},\\ {\mathrm{\Gamma }}_4& =\frac{\mathrm{\partial }}{\mathrm{\partial \Psi }},{\mathrm{\Gamma }}_5=F^1(\mathrm{ϵh}+f-g)\frac{\mathrm{\partial }}{\mathrm{\partial \Psi }},\\ {\mathrm{\Gamma }}_6& =\theta {ϵ}^{2}g\frac{\mathrm{\partial }}{\mathrm{\partial }g}+\mathrm{\theta ϵ}(\mathrm{ϵh}+f)\frac{\mathrm{\partial }}{\mathrm{\partial }h}+\mathrm{\gamma g}\frac{\mathrm{\partial }}{\mathrm{\partial \Psi }},\\ {\mathrm{\Gamma }}_7& =(\alpha -\text{β})\mathrm{ϵf}\frac{\mathrm{\partial }}{\mathrm{\partial }f}+ϵ(2\mathrm{ϵ\alpha h}+2\mathrm{\alpha f}-\mathrm{\alpha g}-\text{β}g)\frac{\mathrm{\partial }}{\mathrm{\partial }g}\\ & +(\mathrm{ϵ\alpha h}+\text{β}\mathrm{ϵh}+2\text{β}f-2\text{β}g)\frac{\mathrm{\partial }}{\mathrm{\partial }h}\\ & +(\alpha -\text{β})ϵ\mathrm{\Psi }\frac{\mathrm{\partial }}{\mathrm{\partial \Psi }},\\ {\mathrm{\Gamma }}_8& =2\theta {ϵ}^2(\alpha -\text{β})f\frac{\mathrm{\partial }}{\mathrm{\partial }f}+2\mathrm{\alpha \theta }{ϵ}^2(\mathrm{ϵh}+f-g)\frac{\mathrm{\partial }}{\mathrm{\partial }g}\\ & -2\mathrm{\theta ϵ}(\mathrm{\alpha f}-\text{β}\mathrm{ϵh}-2\text{β}f+\text{β}g)\frac{\mathrm{\partial }}{\mathrm{\partial }h}\\ & +\gamma (\alpha -\text{β})f\frac{\mathrm{\partial }}{\mathrm{\partial \Psi }},\end{array}$ with $F^1$ serving as an arbitrary functions of f, g and h. We combine linearly the three translational generators ${\mathrm{\Gamma }}_1=\mathrm{\partial }/\mathrm{\partial f}$, ${\mathrm{\Gamma }}_2=\mathrm{\partial }/\mathrm{\partial g}$ and ${\mathrm{\Gamma }}_3=\mathrm{\partial }/\mathrm{\partial h}$ as $\mathrm{\Gamma }={\mathrm{\Gamma }}_1+{\mathrm{\Gamma }}_2+{\mathrm{\Gamma }}_3$ to reduce Equation (58(58) $\begin{array}{rl}& \alpha {ϵ}^4{\mathrm{\Psi }}_{\mathrm{fffg}}-2\theta {ϵ}^3{\mathrm{\Psi }}_f{\mathrm{\Psi }}_{\mathrm{fg}}-\theta {ϵ}^3{\mathrm{\Psi }}_g{\mathrm{\Psi }}_{\mathrm{ff}}-3\alpha {ϵ}^3{\mathrm{\Psi }}_{\mathrm{ffgh}}\\ & +\text{β}{ϵ}^3{\mathrm{\Psi }}_{\mathrm{fffh}}+2\theta {ϵ}^2{\mathrm{\Psi }}_{\mathrm{gh}}{\mathrm{\Psi }}_f\\ & +2\theta {ϵ}^2{\mathrm{\Psi }}_g{\mathrm{\Psi }}_{\mathrm{fh}}-2\theta {ϵ}^2{\mathrm{\Psi }}_f{\mathrm{\Psi }}_{\mathrm{fh}}+2\theta {ϵ}^2{\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{fg}}\\ & +3\alpha {ϵ}^2{\mathrm{\Psi }}_{\mathrm{fghh}}-\theta {ϵ}^2{\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{ff}}-3\text{β}{ϵ}^2{\mathrm{\Psi }}_{\mathrm{ffhh}}\\ & -\mathrm{\theta ϵ}{\mathrm{\Psi }}_{\mathrm{hh}}{\mathrm{\Psi }}_g-2\mathrm{\theta ϵ}{\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{gh}}-\mathrm{\alpha ϵ}{\mathrm{\Psi }}_{\mathrm{ghhh}}\\ & +2\mathrm{\theta ϵ}{\mathrm{\Psi }}_{\mathrm{hh}}{\mathrm{\Psi }}_f+4\mathrm{\theta ϵ}{\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{fh}}+3\text{β}ϵ{\mathrm{\Psi }}_{\mathrm{fhhh}}+\mathrm{\gamma ϵ}{\mathrm{\Psi }}_{\mathrm{fg}}\\ & +\mathrm{\gamma ϵ}{\mathrm{\Psi }}_{\mathrm{ff}}-3\theta {\mathrm{\Psi }}_h{\mathrm{\Psi }}_{\mathrm{hh}}-\text{β}{\mathrm{\Psi }}_{\mathrm{hhhh}}\\ & -\gamma {\mathrm{\Psi }}_{\mathrm{gh}}-\gamma {\mathrm{\Psi }}_{\mathrm{fh}}=0.\end{array}$(58) ). Solution to the Lagrangian system related to Γ as given before leads to the invariants (59) $s=g-h,r=f-g,\psi =\mathrm{\Psi }.$(59) Making ψ the new dependent variable along with r as well as s as the latest independent variables, 3D-genBSMe (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) becomes (60) $\begin{array}{rl}& \alpha {ϵ}^4{\psi }_{\mathrm{rrrs}}-\alpha {ϵ}^4{\psi }_{\mathrm{rrrr}}+3\alpha {ϵ}^3{\psi }_{\mathrm{rrss}}-3\alpha {ϵ}^3{\psi }_{\mathrm{rrrs}}\\ & +3\alpha {ϵ}^2{\psi }_{\mathrm{rsss}}-3\alpha {ϵ}^2{\psi }_{\mathrm{rrss}}\\ & +\mathrm{\alpha ϵ}{\psi }_{\mathrm{ssss}}-\mathrm{\alpha ϵ}{\psi }_{\mathrm{rsss}}-\text{β}{ϵ}^3{\psi }_{\mathrm{rrrs}}-3\text{β}{ϵ}^2{\psi }_{\mathrm{rrss}}\\ & -3\text{β}ϵ{\psi }_{\mathrm{rsss}}-\text{β}{\psi }_{\mathrm{ssss}}+\mathrm{\gamma ϵ}{\psi }_{\mathrm{rs}}\\ & +\gamma {\psi }_{\mathrm{ss}}+3\theta {\psi }_s{\psi }_{\mathrm{ss}}-2\theta {ϵ}^3{\psi }_r{\psi }_{\mathrm{rs}}-\theta {ϵ}^3{\psi }_s{\psi }_{\mathrm{rr}}\\ & +3\theta {ϵ}^3{\psi }_r{\psi }_{\mathrm{rr}}-2\theta {ϵ}^2{\psi }_{\mathrm{ss}}{\psi }_r\\ & -4\theta {ϵ}^2{\psi }_s{\psi }_{\mathrm{rs}}+6\theta {ϵ}^2{\psi }_r{\psi }_{\mathrm{rs}}+3\theta {ϵ}^2{\psi }_s{\psi }_{\mathrm{rr}}\\ & -3\mathrm{\theta ϵ}{\psi }_s{\psi }_{\mathrm{ss}}+3\mathrm{\theta ϵ}{\psi }_{\mathrm{ss}}{\psi }_r+6\mathrm{\theta ϵ}{\psi }_s{\psi }_{\mathrm{rs}}=0,\end{array}$(60) which is evidently an NLPDE in terms of two independent variables. One invokes the symmetries of (60(60) $\begin{array}{rl}& \alpha {ϵ}^4{\psi }_{\mathrm{rrrs}}-\alpha {ϵ}^4{\psi }_{\mathrm{rrrr}}+3\alpha {ϵ}^3{\psi }_{\mathrm{rrss}}-3\alpha {ϵ}^3{\psi }_{\mathrm{rrrs}}\\ & +3\alpha {ϵ}^2{\psi }_{\mathrm{rsss}}-3\alpha {ϵ}^2{\psi }_{\mathrm{rrss}}\\ & +\mathrm{\alpha ϵ}{\psi }_{\mathrm{ssss}}-\mathrm{\alpha ϵ}{\psi }_{\mathrm{rsss}}-\text{β}{ϵ}^3{\psi }_{\mathrm{rrrs}}-3\text{β}{ϵ}^2{\psi }_{\mathrm{rrss}}\\ & -3\text{β}ϵ{\psi }_{\mathrm{rsss}}-\text{β}{\psi }_{\mathrm{ssss}}+\mathrm{\gamma ϵ}{\psi }_{\mathrm{rs}}\\ & +\gamma {\psi }_{\mathrm{ss}}+3\theta {\psi }_s{\psi }_{\mathrm{ss}}-2\theta {ϵ}^3{\psi }_r{\psi }_{\mathrm{rs}}-\theta {ϵ}^3{\psi }_s{\psi }_{\mathrm{rr}}\\ & +3\theta {ϵ}^3{\psi }_r{\psi }_{\mathrm{rr}}-2\theta {ϵ}^2{\psi }_{\mathrm{ss}}{\psi }_r\\ & -4\theta {ϵ}^2{\psi }_s{\psi }_{\mathrm{rs}}+6\theta {ϵ}^2{\psi }_r{\psi }_{\mathrm{rs}}+3\theta {ϵ}^2{\psi }_s{\psi }_{\mathrm{rr}}\\ & -3\mathrm{\theta ϵ}{\psi }_s{\psi }_{\mathrm{ss}}+3\mathrm{\theta ϵ}{\psi }_{\mathrm{ss}}{\psi }_r+6\mathrm{\theta ϵ}{\psi }_s{\psi }_{\mathrm{rs}}=0,\end{array}$(60) ) to change the said equation to an ODE. Equation (60(60) $\begin{array}{rl}& \alpha {ϵ}^4{\psi }_{\mathrm{rrrs}}-\alpha {ϵ}^4{\psi }_{\mathrm{rrrr}}+3\alpha {ϵ}^3{\psi }_{\mathrm{rrss}}-3\alpha {ϵ}^3{\psi }_{\mathrm{rrrs}}\\ & +3\alpha {ϵ}^2{\psi }_{\mathrm{rsss}}-3\alpha {ϵ}^2{\psi }_{\mathrm{rrss}}\\ & +\mathrm{\alpha ϵ}{\psi }_{\mathrm{ssss}}-\mathrm{\alpha ϵ}{\psi }_{\mathrm{rsss}}-\text{β}{ϵ}^3{\psi }_{\mathrm{rrrs}}-3\text{β}{ϵ}^2{\psi }_{\mathrm{rrss}}\\ & -3\text{β}ϵ{\psi }_{\mathrm{rsss}}-\text{β}{\psi }_{\mathrm{ssss}}+\mathrm{\gamma ϵ}{\psi }_{\mathrm{rs}}\\ & +\gamma {\psi }_{\mathrm{ss}}+3\theta {\psi }_s{\psi }_{\mathrm{ss}}-2\theta {ϵ}^3{\psi }_r{\psi }_{\mathrm{rs}}-\theta {ϵ}^3{\psi }_s{\psi }_{\mathrm{rr}}\\ & +3\theta {ϵ}^3{\psi }_r{\psi }_{\mathrm{rr}}-2\theta {ϵ}^2{\psi }_{\mathrm{ss}}{\psi }_r\\ & -4\theta {ϵ}^2{\psi }_s{\psi }_{\mathrm{rs}}+6\theta {ϵ}^2{\psi }_r{\psi }_{\mathrm{rs}}+3\theta {ϵ}^2{\psi }_s{\psi }_{\mathrm{rr}}\\ & -3\mathrm{\theta ϵ}{\psi }_s{\psi }_{\mathrm{ss}}+3\mathrm{\theta ϵ}{\psi }_{\mathrm{ss}}{\psi }_r+6\mathrm{\theta ϵ}{\psi }_s{\psi }_{\mathrm{rs}}=0,\end{array}$(60) ) yields the symmetries $\begin{array}{rl}{M}_1& =\frac{\mathrm{\partial }}{\mathrm{\partial }r},M_2=\frac{\mathrm{\partial }}{\mathrm{\partial }s},M_3=\frac{\mathrm{\partial }}{\mathrm{\partial }\psi },\\ M_4& =\theta {ϵ}^{3}r\frac{\mathrm{\partial }}{\mathrm{\partial }r}+\theta {ϵ}^{3}s\frac{\mathrm{\partial }}{\mathrm{\partial }r}\\ & -(\theta {ϵ}^3\psi -\mathrm{ϵ\gamma r}-3\mathrm{ϵ\gamma s}+3\mathrm{\gamma r})\frac{\mathrm{\partial }}{\mathrm{\partial }\psi }.\end{array}$ Combining translation symmetries $M_1$ and $M_2$ in the structure $M=\nu M_1+M_2$, with non-zero real constant ν, and adopting the action earlier taken yields the invariants (61) $\zeta =r-\mathrm{\nu s}\text{and}\mathrm{\Theta }=\psi .$(61) The use of group invariant solution $\psi =\mathrm{\Theta }\left(\zeta \right)$ makes it possible for Equation (60(60) $\begin{array}{rl}& \alpha {ϵ}^4{\psi }_{\mathrm{rrrs}}-\alpha {ϵ}^4{\psi }_{\mathrm{rrrr}}+3\alpha {ϵ}^3{\psi }_{\mathrm{rrss}}-3\alpha {ϵ}^3{\psi }_{\mathrm{rrrs}}\\ & +3\alpha {ϵ}^2{\psi }_{\mathrm{rsss}}-3\alpha {ϵ}^2{\psi }_{\mathrm{rrss}}\\ & +\mathrm{\alpha ϵ}{\psi }_{\mathrm{ssss}}-\mathrm{\alpha ϵ}{\psi }_{\mathrm{rsss}}-\text{β}{ϵ}^3{\psi }_{\mathrm{rrrs}}-3\text{β}{ϵ}^2{\psi }_{\mathrm{rrss}}\\ & -3\text{β}ϵ{\psi }_{\mathrm{rsss}}-\text{β}{\psi }_{\mathrm{ssss}}+\mathrm{\gamma ϵ}{\psi }_{\mathrm{rs}}\\ & +\gamma {\psi }_{\mathrm{ss}}+3\theta {\psi }_s{\psi }_{\mathrm{ss}}-2\theta {ϵ}^3{\psi }_r{\psi }_{\mathrm{rs}}-\theta {ϵ}^3{\psi }_s{\psi }_{\mathrm{rr}}\\ & +3\theta {ϵ}^3{\psi }_r{\psi }_{\mathrm{rr}}-2\theta {ϵ}^2{\psi }_{\mathrm{ss}}{\psi }_r\\ & -4\theta {ϵ}^2{\psi }_s{\psi }_{\mathrm{rs}}+6\theta {ϵ}^2{\psi }_r{\psi }_{\mathrm{rs}}+3\theta {ϵ}^2{\psi }_s{\psi }_{\mathrm{rr}}\\ & -3\mathrm{\theta ϵ}{\psi }_s{\psi }_{\mathrm{ss}}+3\mathrm{\theta ϵ}{\psi }_{\mathrm{ss}}{\psi }_r+6\mathrm{\theta ϵ}{\psi }_s{\psi }_{\mathrm{rs}}=0,\end{array}$(60) ) to be transformed to a new fourth-order NLODE expressed as (62) $\begin{array}{rl}& 2{\vartheta }^3(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1\left)\right){\mathrm{\Theta }}^{\mathrm{\prime \prime \prime \prime }}\left(\zeta \right)\\ & +6\theta {\vartheta }^2\left(\nu \right(ϵ-1)+ϵ){\mathrm{\Theta }}^{\mathrm{\prime }}\left(\zeta \right){\mathrm{\Theta }}^{\mathrm{\prime \prime }}\left(\zeta \right)\\ & -\mathrm{\gamma \nu }{\xi }_0{\mathrm{\Theta }}^{\mathrm{\prime \prime }}\left(\zeta \right)=0,\end{array}$(62) with ${\xi }_0=2\vartheta$, $\vartheta =(ϵ-\nu )$, along with $\zeta =(\nu -ϵ)x+ϵ(\nu +1)y-\mathrm{\nu z}-\mathrm{\nu t}$.

Direct integration and solution of NLODE (62(62) $\begin{array}{rl}& 2{\vartheta }^3(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1\left)\right){\mathrm{\Theta }}^{\mathrm{\prime \prime \prime \prime }}\left(\zeta \right)\\ & +6\theta {\vartheta }^2\left(\nu \right(ϵ-1)+ϵ){\mathrm{\Theta }}^{\mathrm{\prime }}\left(\zeta \right){\mathrm{\Theta }}^{\mathrm{\prime \prime }}\left(\zeta \right)\\ & -\mathrm{\gamma \nu }{\xi }_0{\mathrm{\Theta }}^{\mathrm{\prime \prime }}\left(\zeta \right)=0,\end{array}$(62) )

Elliptic integral solution of the second kind

Integration of (62(62) $\begin{array}{rl}& 2{\vartheta }^3(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1\left)\right){\mathrm{\Theta }}^{\mathrm{\prime \prime \prime \prime }}\left(\zeta \right)\\ & +6\theta {\vartheta }^2\left(\nu \right(ϵ-1)+ϵ){\mathrm{\Theta }}^{\mathrm{\prime }}\left(\zeta \right){\mathrm{\Theta }}^{\mathrm{\prime \prime }}\left(\zeta \right)\\ & -\mathrm{\gamma \nu }{\xi }_0{\mathrm{\Theta }}^{\mathrm{\prime \prime }}\left(\zeta \right)=0,\end{array}$(62) ) once with regard to ζ purveys (63) $\begin{array}{rl}& 2{\vartheta }^3(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1\left)\right){\mathrm{\Theta }}^{\mathrm{\prime \prime \prime }}\left(\zeta \right)\\ & +3\theta {\vartheta }^2\left(\nu \right(ϵ-1)+ϵ){\mathrm{\Theta }}^{\mathrm{\prime }2}\left(\zeta \right)\\ & -\mathrm{\gamma \nu }{\xi }_0{\mathrm{\Theta }}^{\mathrm{\prime }}\left(\zeta \right)+B_0=0,\end{array}$(63) with $B_0$, regarded as an integration constant. Using the assumption that $\begin{array}{rl}{\mathrm{\Theta }}^{\mathrm{\prime }}\left(\zeta \right)& =\left(\frac{2(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1\left)\right)}{\theta \left(\nu \right(ϵ-1)+ϵ)}\right)\vartheta \mathrm{\Phi }\left(\zeta \right),\\ \mu & =\frac{\mathrm{\gamma \nu }{\xi }_0}{2{\vartheta }^3(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1\left)\right)},\\ B_1& =\frac{\theta B_0\left(\nu \right(ϵ-1)+ϵ)}{4{\vartheta }^3(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1){)}^2}\end{array}$ thus making Equation (63(63) $\begin{array}{rl}& 2{\vartheta }^3(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1\left)\right){\mathrm{\Theta }}^{\mathrm{\prime \prime \prime }}\left(\zeta \right)\\ & +3\theta {\vartheta }^2\left(\nu \right(ϵ-1)+ϵ){\mathrm{\Theta }}^{\mathrm{\prime }2}\left(\zeta \right)\\ & -\mathrm{\gamma \nu }{\xi }_0{\mathrm{\Theta }}^{\mathrm{\prime }}\left(\zeta \right)+B_0=0,\end{array}$(63) ) becoming an ODE of second order, viz. (64) ${\mathrm{\Phi }}^{\mathrm{\prime \prime }}\left(\zeta \right)+3\mathrm{\Phi }(\zeta {)}^2-\mu \mathrm{\Phi }(\zeta )+B_1=0,$(64) which one can integrate without much stress. Multiplication of (64(64) ${\mathrm{\Phi }}^{\mathrm{\prime \prime }}\left(\zeta \right)+3\mathrm{\Phi }(\zeta {)}^2-\mu \mathrm{\Phi }(\zeta )+B_1=0,$(64) ) by ${\mathrm{\Phi }}^{\mathrm{\prime }}\left(\zeta \right)$ and later integrating the outcome once with regard to ζ, we secure (65) ${\mathrm{\Phi }}^{\mathrm{\prime }}(\zeta {)}^2=-(2\mathrm{\Phi }(\zeta {)}^3-\mu \mathrm{\Phi }(\zeta {)}^2+2B_1\mathrm{\Phi }+2B_2),$(65) where $B_2$ is taken as the integration constant. Suppose that we have the cubic equation $\mathrm{\Phi }(\zeta {)}^3-\frac{1}{2}\mu \mathrm{\Phi }(\zeta {)}^2+B_1\mathrm{\Phi }\left(\zeta \right)+B_2=0$ possessive of the roots ${\text{β}}_3<{\text{β}}_2<{\text{β}}_1$, where ${\text{β}}_1$, ${\text{β}}_2$ and ${\text{β}}_3$ are real numbers, thus we get (66) ${\mathrm{\Phi }}^{\mathrm{\prime }}(\zeta {)}^2=-2(\mathrm{\Phi }\left(\zeta \right)-{\text{β}}_1\left)\right(\mathrm{\Phi }\left(\zeta \right)-{\text{β}}_2\left)\right(\mathrm{\Phi }\left(\zeta \right)-{\text{β}}_3).$(66) Equation (66(66) ${\mathrm{\Phi }}^{\mathrm{\prime }}(\zeta {)}^2=-2(\mathrm{\Phi }\left(\zeta \right)-{\text{β}}_1\left)\right(\mathrm{\Phi }\left(\zeta \right)-{\text{β}}_2\left)\right(\mathrm{\Phi }\left(\zeta \right)-{\text{β}}_3).$(66) ) furnishes the Jacobi-elliptic-cosine function solution (JCECFS), designated as [11] (67) $\begin{array}{rl}\mathrm{\Phi }\left(\zeta \right)& ={\text{β}}_2+({\text{β}}_1-{\text{β}}_2){\mathrm{cn}}^2\left(\sqrt{\frac{{\text{β}}_1-{\text{β}}_3}{2}}\zeta |Q^2\right),\\ Q^2& =\frac{{\text{β}}_1-{\text{β}}_2}{{\text{β}}_1-{\text{β}}_3}.\end{array}$(67) In this instance, we have $\mathrm{cn}\left(\zeta |Q^2\right)$ referring to the Jacobi cosine elliptic function. In addition, the solution of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) in this regard can be ultimately expressed as $\mathrm{\Theta }\left(\zeta \right)=\frac{2(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1\left)\right)}{\theta \left(\nu \right(ϵ-1)+ϵ)}\int \mathrm{\Phi }\left(\zeta \right)\mathrm{d}\zeta .$ Furthermore, it is noteworthy and germane to state categorically assert here that special limits taken regarding the JCECFS, one could achieve diverse functions. Take for instance $\underset{Q^2\to 0}{lim}\mathrm{cn}\left(\zeta |Q^2\right)=\cos \left(\zeta \right)$ and $\underset{Q^2\to 1}{lim}\mathrm{cn}\left(\zeta |Q^2\right)=\mathrm{sech}\left(\zeta \right)$ [70]. On retrograding to the usual variables, one secures a general analytic solution of 3D-genBSMe (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) as (68) $\begin{array}{rl}& u(t,x,y,z)\\ & =\frac{2(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1\left)\right)}{\theta \left(\nu \right(ϵ-1)+ϵ)}\{\phantom{\frac{({\text{β}}_1-{\text{β}}_2)\mathrm{dn}\left(B\zeta |Q^2\right)E\left(\mathrm{am}\right(B\zeta |Q^2\left)|Q^2\right)}{B{Q}^2\sqrt{\mathrm{dn}(B\zeta |Q^2{)}^2}}}\frac{\{{\text{β}}_2-(1-Q^2\left){\text{β}}_1\right\}\zeta }{Q^2}\\ & +\frac{({\text{β}}_1-{\text{β}}_2)\mathrm{dn}\left(B\zeta |Q^2\right)E\left(\mathrm{am}\right(B\zeta |Q^2\left)|Q^2\right)}{B{Q}^2\sqrt{\mathrm{dn}(B\zeta |Q^2{)}^2}}\},\end{array}$(68) with $\zeta =(\nu -ϵ)x+ϵ(\nu +1)y-\mathrm{\nu z}-\mathrm{\nu t}$ and $E\left(\zeta |Q^2\right)$ depicting the second-kind-elliptic integral function relation. Additionally, am$\left(\zeta |Q^2\right)$ stands for the amplitude function, whereas dn$\left(\zeta |Q^2\right)$ connotes delta amplitude function with cn$\left(\zeta |Q^2\right)$ denoting the Jacobi cosine function and also $B=\sqrt{({\text{β}}_1-{\text{β}}_3)}/\sqrt{2}$ [70,71]. The dynamical behaviour of solution (68(68) $\begin{array}{rl}& u(t,x,y,z)\\ & =\frac{2(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1\left)\right)}{\theta \left(\nu \right(ϵ-1)+ϵ)}\{\phantom{\frac{({\text{β}}_1-{\text{β}}_2)\mathrm{dn}\left(B\zeta |Q^2\right)E\left(\mathrm{am}\right(B\zeta |Q^2\left)|Q^2\right)}{B{Q}^2\sqrt{\mathrm{dn}(B\zeta |Q^2{)}^2}}}\frac{\{{\text{β}}_2-(1-Q^2\left){\text{β}}_1\right\}\zeta }{Q^2}\\ & +\frac{({\text{β}}_1-{\text{β}}_2)\mathrm{dn}\left(B\zeta |Q^2\right)E\left(\mathrm{am}\right(B\zeta |Q^2\left)|Q^2\right)}{B{Q}^2\sqrt{\mathrm{dn}(B\zeta |Q^2{)}^2}}\},\end{array}$(68) ) is displayed with diverse values of $(t,x,y)$ in Figures 3 and 4.

The simplest equation technique (SET)

We invoke in this section of the study, SET to gain some closed-form solutions of the third-order ODE (62(62) $\begin{array}{rl}& 2{\vartheta }^3(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1\left)\right){\mathrm{\Theta }}^{\mathrm{\prime \prime \prime \prime }}\left(\zeta \right)\\ & +6\theta {\vartheta }^2\left(\nu \right(ϵ-1)+ϵ){\mathrm{\Theta }}^{\mathrm{\prime }}\left(\zeta \right){\mathrm{\Theta }}^{\mathrm{\prime \prime }}\left(\zeta \right)\\ & -\mathrm{\gamma \nu }{\xi }_0{\mathrm{\Theta }}^{\mathrm{\prime \prime }}\left(\zeta \right)=0,\end{array}$(62) ). This approach was established by Kudryashov [39] and subsequently Vitanov made an amend to it [72]. Here simplest equation to be invoked is stated as the Bernoulli as well as Riccati equations. Solution to the NLODEs can be explicated with regard to elementary functions. Let us contemplate the solution of Equation (62(62) $\begin{array}{rl}& 2{\vartheta }^3(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1\left)\right){\mathrm{\Theta }}^{\mathrm{\prime \prime \prime \prime }}\left(\zeta \right)\\ & +6\theta {\vartheta }^2\left(\nu \right(ϵ-1)+ϵ){\mathrm{\Theta }}^{\mathrm{\prime }}\left(\zeta \right){\mathrm{\Theta }}^{\mathrm{\prime \prime }}\left(\zeta \right)\\ & -\mathrm{\gamma \nu }{\xi }_0{\mathrm{\Theta }}^{\mathrm{\prime \prime }}\left(\zeta \right)=0,\end{array}$(62) ) in the structure (69) $\mathrm{\Theta }\left(\zeta \right)=\sum _{j=0}^M{A}_j\left(U\right(\zeta ){)}^j,$(69) with $U\left(\zeta \right)$ satisfying the earlier-mentioned NLODEs. Real-positive integer M can be decided via balancing steps [73]. In addition, $A_0,\dots ,A_M$ are parameters which we have to determine.

One observes that the Bernoulli NLODE (70) $U^{\mathrm{\prime }}\left(\zeta \right)=\mathrm{aU}\left(\zeta \right)+b{U}^2\left(\zeta \right),$(70) whose solutions are explicated as (71) $\begin{array}{rl}U\left(\zeta \right)& =\frac{-a{Q}_0}{b\{Q_0+\cosh \left[a(\zeta +Q_1)\right]-\sinh \left[a(\zeta +Q_1)\right]\}},\end{array}$(71) (72) $\begin{array}{rl}U\left(\zeta \right)& =a\left\{\frac{\cosh \left[a\right(\zeta +K\left)\right]+\sinh \left[a\right(\zeta +K\left)\right]}{1-\mathrm{bcosh}\left[a\right(\zeta +K\left)\right]-\mathrm{bsinh}\left[a\right(\zeta +K\left)\right]}\right\}.\end{array}$(72) In the case of Riccati NLODE expressed as (73) $U^{\mathrm{\prime }}\left(\zeta \right)=a{U}^2\left(\zeta \right)+\mathrm{bU}\left(\zeta \right)+c,$(73) we shall be using the solutions $U\left(\zeta \right)=-\frac{b}{2a}-\frac{\varphi }{2a}\tanh \left[\frac{1}{2}\varphi \right(\zeta +K\left)\right]$ alongside $\begin{array}{rl}U\left(\zeta \right)& =-\frac{b}{2a}-\frac{\varphi }{2a}\tanh \left(\frac{1}{2}\mathrm{\varphi \zeta }\right)\\ & +\frac{\mathrm{sech}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)}{\mathrm{Kcosh}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)-\frac{2a}{\varphi }\sinh \left(\frac{\mathrm{\varphi \zeta }}{2}\right)},\end{array}$ with ${\varphi }^2=b^2-4\mathrm{ac}>0$ and $Q_0$, $Q_1$, K, representing arbitrary constants.

Solutions of (62(62) $\begin{array}{rl}& 2{\vartheta }^3(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1\left)\right){\mathrm{\Theta }}^{\mathrm{\prime \prime \prime \prime }}\left(\zeta \right)\\ & +6\theta {\vartheta }^2\left(\nu \right(ϵ-1)+ϵ){\mathrm{\Theta }}^{\mathrm{\prime }}\left(\zeta \right){\mathrm{\Theta }}^{\mathrm{\prime \prime }}\left(\zeta \right)\\ & -\mathrm{\gamma \nu }{\xi }_0{\mathrm{\Theta }}^{\mathrm{\prime \prime }}\left(\zeta \right)=0,\end{array}$(62) ) using Bernoulli equation as the simplest equation

The balancing procedure produces M = 1, thus (62(62) $\begin{array}{rl}& 2{\vartheta }^3(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1\left)\right){\mathrm{\Theta }}^{\mathrm{\prime \prime \prime \prime }}\left(\zeta \right)\\ & +6\theta {\vartheta }^2\left(\nu \right(ϵ-1)+ϵ){\mathrm{\Theta }}^{\mathrm{\prime }}\left(\zeta \right){\mathrm{\Theta }}^{\mathrm{\prime \prime }}\left(\zeta \right)\\ & -\mathrm{\gamma \nu }{\xi }_0{\mathrm{\Theta }}^{\mathrm{\prime \prime }}\left(\zeta \right)=0,\end{array}$(62) ) assumes the structure (74) $\mathrm{\Theta }\left(\zeta \right)=A_0+A_{1}U.$(74) Invoking the value of Θ expressed in (74(74) $\mathrm{\Theta }\left(\zeta \right)=A_0+A_{1}U.$(74) ) into NLODE (62(62) $\begin{array}{rl}& 2{\vartheta }^3(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1\left)\right){\mathrm{\Theta }}^{\mathrm{\prime \prime \prime \prime }}\left(\zeta \right)\\ & +6\theta {\vartheta }^2\left(\nu \right(ϵ-1)+ϵ){\mathrm{\Theta }}^{\mathrm{\prime }}\left(\zeta \right){\mathrm{\Theta }}^{\mathrm{\prime \prime }}\left(\zeta \right)\\ & -\mathrm{\gamma \nu }{\xi }_0{\mathrm{\Theta }}^{\mathrm{\prime \prime }}\left(\zeta \right)=0,\end{array}$(62) ), utilizing the Bernoulli (70(70) $U^{\mathrm{\prime }}\left(\zeta \right)=\mathrm{aU}\left(\zeta \right)+b{U}^2\left(\zeta \right),$(70) ) and following the earlier given steps, one secures five long system of algebraic equations with regard to $A_0$ and $A_1$, presented as $\begin{array}{rl}& \alpha a^4\nu {ϵ}^4{A}_1+\alpha a^4{ϵ}^4{A}_1-3\alpha a^4{\nu }^2{ϵ}^3{A}_1-3\alpha a^4\nu {ϵ}^3{A}_1\\ & +3\alpha a^4{\nu }^3{ϵ}^2{A}_1+3\alpha a^4{\nu }^2{ϵ}^2{A}_1\\ & -\alpha a^4{\nu }^4ϵA_1-\alpha a^4{\nu }^3ϵA_1+a^4\text{β}{\nu }^4{A}_1-a^4\text{β}\nu {ϵ}^3{A}_1\\ & +3a^4\text{β}{\nu }^2{ϵ}^2{A}_1-3a^4\text{β}{\nu }^3ϵA_1\\ & -a^2\gamma {\nu }^2{A}_1+a^2\mathrm{\gamma \nu ϵ}{A}_1=0,\\ & 15a^3\mathrm{\alpha b\nu }{ϵ}^4{A}_1+15a^3\mathrm{\alpha b}{ϵ}^4{A}_1-45a^3\mathrm{\alpha b}{\nu }^2{ϵ}^3{A}_1\\ & -45a^3\mathrm{\alpha b\nu }{ϵ}^3{A}_1+45a^3\mathrm{\alpha b}{\nu }^3{ϵ}^2{A}_1\\ & +45a^3\mathrm{\alpha b}{\nu }^2{ϵ}^2{A}_1-15a^3\mathrm{\alpha b}{\nu }^4ϵA_1-15a^3\mathrm{\alpha b}{\nu }^3ϵA_1\\ & +15a^{3}b\text{β}{\nu }^4{A}_1-15a^{3}b\text{β}\nu {ϵ}^3{A}_1\\ & +45a^{3}b\text{β}{\nu }^2{ϵ}^2{A}_1-45a^{3}b\text{β}{\nu }^3ϵA_1\\ & +3a^3\theta {\nu }^3{A}_1^2-3a^3\mathrm{\theta \nu }{ϵ}^3{A}_1^2-3a^3\theta {ϵ}^3{A}_1^2\\ & +6a^3\theta {\nu }^2{ϵ}^2{A}_1^2+9a^3\mathrm{\theta \nu }{ϵ}^2{A}_1^2-3a^3\theta {\nu }^3ϵA_1^2\\ & -9a^3\theta {\nu }^2ϵA_1^2-3\mathrm{ab\gamma }{\nu }^2{A}_1+3\mathrm{ab\gamma \nu ϵ}{A}_1=0,\end{array}$ $\begin{array}{rl}& 50a^2\alpha b^2\nu {ϵ}^4{A}_1+50a^2\alpha b^2{ϵ}^4{A}_1-150a^2\alpha b^2{\nu }^2{ϵ}^3{A}_1\\ & -150a^2\alpha b^2\nu {ϵ}^3{A}_1\\ & +150a^2\alpha b^2{\nu }^3{ϵ}^2{A}_1+150a^2\alpha b^2{\nu }^2{ϵ}^2{A}_1\\ & -50a^2\alpha b^2{\nu }^4ϵA_1-50a^2\alpha b^2{\nu }^3ϵA_1\\ & +50a^2{b}^2\text{β}{\nu }^4{A}_1-50a^2{b}^2\text{β}\nu {ϵ}^3{A}_1\\ & +150a^2{b}^2\text{β}{\nu }^2{ϵ}^2{A}_1-150a^2{b}^2\text{β}{\nu }^3ϵA_1\\ & +12a^2\mathrm{b\theta }{\nu }^3{A}_1^2-12a^2\mathrm{b\theta \nu }{ϵ}^3{A}_1^2-12a^2\mathrm{b\theta }{ϵ}^3{A}_1^2\\ & +24a^2\mathrm{b\theta }{\nu }^2{ϵ}^2{A}_1^2+36a^2\mathrm{b\theta \nu }{ϵ}^2{A}_1^2\\ & -12a^2\mathrm{b\theta }{\nu }^3ϵA_1^2-36a^2\mathrm{b\theta }{\nu }^2ϵA_1^2\\ & -2b^2\gamma {\nu }^2{A}_1+2b^2\mathrm{\gamma \nu ϵ}{A}_1=0,\end{array}$ $\begin{array}{rl}& 60\mathrm{a\alpha }{b}^3\nu {ϵ}^4{A}_1+60\mathrm{a\alpha }{b}^3{ϵ}^4{A}_1-180\mathrm{a\alpha }{b}^3{\nu }^2{ϵ}^3{A}_1\\ & -180\mathrm{a\alpha }{b}^3\nu {ϵ}^3{A}_1+180\mathrm{a\alpha }{b}^3{\nu }^3{ϵ}^2{A}_1\\ & +180\mathrm{a\alpha }{b}^3{\nu }^2{ϵ}^2{A}_1-60\mathrm{a\alpha }{b}^3{\nu }^4ϵA_1-60\mathrm{a\alpha }{b}^3{\nu }^3ϵA_1\\ & +60a{b}^3\text{β}{\nu }^4{A}_1-60a{b}^3\text{β}\nu {ϵ}^3{A}_1\\ & +180a{b}^3\text{β}{\nu }^2{ϵ}^2{A}_1-180a{b}^3\text{β}{\nu }^3ϵA_1\\ & +15a{b}^2\theta {\nu }^3{A}_1^2-15a{b}^2\mathrm{\theta \nu }{ϵ}^3{A}_1^2-15a{b}^2\theta {ϵ}^3{A}_1^2\\ & +30a{b}^2\theta {\nu }^2{ϵ}^2{A}_1^2+45a{b}^2\mathrm{\theta \nu }{ϵ}^2{A}_1^2\\ & -15a{b}^2\theta {\nu }^3ϵA_1^2-45a{b}^2\theta {\nu }^2ϵA_1^2=0,\end{array}$ $\begin{array}{rl}& 24\alpha b^4\nu {ϵ}^4{A}_1+24\alpha b^4{ϵ}^4{A}_1-72\alpha b^4{\nu }^2{ϵ}^3{A}_1\\ & -72\alpha b^4\nu {ϵ}^3{A}_1+72\alpha b^4{\nu }^3{ϵ}^2{A}_1\\ & +72\alpha b^4{\nu }^2{ϵ}^2{A}_1-24\alpha b^4{\nu }^4ϵA_1-24\alpha b^4{\nu }^3ϵA_1\\ & +24\text{β}{b}^4{\nu }^4{A}_1-24\text{β}{b}^4\nu {ϵ}^3{A}_1\\ & +72\text{β}{b}^4{\nu }^2{ϵ}^2{A}_1-72\text{β}{b}^4{\nu }^3ϵA_1\\ & +6b^3\theta {\nu }^3{A}_1^2-6b^3\mathrm{\theta \nu }{ϵ}^3{A}_1^2-6b^3\theta {ϵ}^3{A}_1^2\\ & +12b^3\theta {\nu }^2{ϵ}^2{A}_1^2+18b^3\mathrm{\theta \nu }{ϵ}^2{A}_1^2\\ & -6b^3\theta {\nu }^3ϵA_1^2-18b^3\theta {\nu }^2ϵA_1^2=0.\end{array}$ On solving the achieved system of algebraic equations via the use of Mathematica, one gains (75) $\begin{array}{rl}{A}_0& =A_0,A_1=\frac{4b(ϵ-\nu )(\mathrm{\alpha \nu ϵ}+\mathrm{\alpha ϵ}-\text{β}\nu )}{\theta (ϵ-\nu +\mathrm{\nu ϵ})},\\ a& =\mp \sqrt{\frac{\mathrm{\gamma \nu }}{\begin{array}{c}-\mathrm{\alpha \nu }{ϵ}^3-\alpha {ϵ}^3+2\alpha {\nu }^2{ϵ}^2+2\mathrm{\alpha \nu }{ϵ}^2-\alpha {\nu }^3ϵ\\ -\alpha {\nu }^2ϵ+\text{β}{\nu }^3+\text{β}\nu {ϵ}^2-2\text{β}{\nu }^2ϵ\end{array}}}.\end{array}$(75) Thus the solution of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) is given by the hyperbolic function solution (76) $\begin{array}{rl}& u(t,x,y,z)\\ & =A_0-A_{1}a\left\{\frac{Q_0}{\begin{array}{c}b\{Q_0+\cosh \left[a(\zeta +Q_1)\right]\\ -\sinh \left[a(\zeta +Q_1)\right]\}\end{array}}\right\},\end{array}$(76) (77) $\begin{array}{rl}& u(t,x,y,z)\\ & =A_0+A_{1}a\left\{\frac{\cosh \left[a\right(\zeta +K\left)\right]+\sinh \left[a\right(\zeta +K\left)\right]}{\begin{array}{c}1-b\cosh \left[a\right(\zeta +K\left)\right]\\ -b\sinh \left[a\right(\zeta +K\left)\right]\end{array}}\right\}\end{array}$(77) with $\zeta =(\nu -ϵ)x+ϵ(\nu +1)y-\mathrm{\nu z}-\mathrm{\nu t}$ along with K taken as an integration constant. We display the dynamics of solution (77(77) $\begin{array}{rl}& u(t,x,y,z)\\ & =A_0+A_{1}a\left\{\frac{\cosh \left[a\right(\zeta +K\left)\right]+\sinh \left[a\right(\zeta +K\left)\right]}{\begin{array}{c}1-b\cosh \left[a\right(\zeta +K\left)\right]\\ -b\sinh \left[a\right(\zeta +K\left)\right]\end{array}}\right\}\end{array}$(77) ) in Figures 7 and 8 with various values of t and x.

Solutions of (62(62) $\begin{array}{rl}& 2{\vartheta }^3(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1\left)\right){\mathrm{\Theta }}^{\mathrm{\prime \prime \prime \prime }}\left(\zeta \right)\\ & +6\theta {\vartheta }^2\left(\nu \right(ϵ-1)+ϵ){\mathrm{\Theta }}^{\mathrm{\prime }}\left(\zeta \right){\mathrm{\Theta }}^{\mathrm{\prime \prime }}\left(\zeta \right)\\ & -\mathrm{\gamma \nu }{\xi }_0{\mathrm{\Theta }}^{\mathrm{\prime \prime }}\left(\zeta \right)=0,\end{array}$(62) ) using Riccati equation as the simplest equation

We use the balancing procedure which occasions M = 1 as earlier found, so the solution of (62(62) $\begin{array}{rl}& 2{\vartheta }^3(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1\left)\right){\mathrm{\Theta }}^{\mathrm{\prime \prime \prime \prime }}\left(\zeta \right)\\ & +6\theta {\vartheta }^2\left(\nu \right(ϵ-1)+ϵ){\mathrm{\Theta }}^{\mathrm{\prime }}\left(\zeta \right){\mathrm{\Theta }}^{\mathrm{\prime \prime }}\left(\zeta \right)\\ & -\mathrm{\gamma \nu }{\xi }_0{\mathrm{\Theta }}^{\mathrm{\prime \prime }}\left(\zeta \right)=0,\end{array}$(62) ) is of the form (78) $\mathrm{\Theta }\left(\zeta \right)=A_0+A_{1}U.$(78) Inserting the value of $\mathrm{\Theta }\left(\zeta \right)$ as given in (78(78) $\mathrm{\Theta }\left(\zeta \right)=A_0+A_{1}U.$(78) ) into (62(62) $\begin{array}{rl}& 2{\vartheta }^3(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1\left)\right){\mathrm{\Theta }}^{\mathrm{\prime \prime \prime \prime }}\left(\zeta \right)\\ & +6\theta {\vartheta }^2\left(\nu \right(ϵ-1)+ϵ){\mathrm{\Theta }}^{\mathrm{\prime }}\left(\zeta \right){\mathrm{\Theta }}^{\mathrm{\prime \prime }}\left(\zeta \right)\\ & -\mathrm{\gamma \nu }{\xi }_0{\mathrm{\Theta }}^{\mathrm{\prime \prime }}\left(\zeta \right)=0,\end{array}$(62) ), making use of the Riccati equation (73(73) $U^{\mathrm{\prime }}\left(\zeta \right)=a{U}^2\left(\zeta \right)+\mathrm{bU}\left(\zeta \right)+c,$(73) ) as well as equating all the coefficients of the functions $U^j$ to zero, we secure six correlating algebraic equations in respect of $A_0$ and $A_1$ as earlier demonstrated, which solve via the use of Mathematica to give $\begin{array}{rl}{A}_0& =A_0,A_1=\frac{4b(ϵ-\nu )(\mathrm{\alpha \nu ϵ}+\mathrm{\alpha ϵ}-\text{β}\nu )}{\theta (\mathrm{\nu ϵ}+ϵ-\nu )},\\ a& =\mp \sqrt{\frac{\begin{array}{c}4\mathrm{\alpha bd\nu }{ϵ}^3+4\mathrm{\alpha bd}{ϵ}^3-8\mathrm{\alpha bd}{\nu }^2{ϵ}^2-8\mathrm{\alpha bd\nu }{ϵ}^2\\ +4\mathrm{\alpha bd}{\nu }^3ϵ+4\mathrm{\alpha bd}{\nu }^2ϵ+{\Theta }_0\end{array}}{\begin{array}{c}\mathrm{\alpha \nu }{ϵ}^3+\alpha {ϵ}^3-2\alpha {\nu }^2{ϵ}^2-2\mathrm{\alpha \nu }{ϵ}^2+\alpha {\nu }^3ϵ\\ +\alpha {\nu }^2ϵ-\text{β}{\nu }^3-\text{β}\nu {ϵ}^2+2\text{β}{\nu }^2ϵ\end{array}}},\end{array}$ where ${\Theta }_0=-4b\text{β}d{\nu }^3-4b\text{β}\mathrm{d\nu }{ϵ}^2+8b\text{β}d{\nu }^2ϵ-\mathrm{\gamma \nu }$ and $\zeta =(\nu -ϵ)x+ϵ(\nu +1)y-\mathrm{\nu z}-\mathrm{\nu t}$. Hence the general solutions of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) can be expressed as (79) $u(t,x,y,z)=A_0+A_1(-\frac{b}{2a}-\frac{\varphi }{2a}\tanh \left[\frac{1}{2}\varphi \right(\zeta +K\left)\right])$(79) and (80) $\begin{array}{rl}u(t,x,y,z)& =A_0+A_1(\phantom{\frac{\mathrm{sech}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)}{\mathrm{Kcosh}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)-\frac{2a}{\varphi }\sinh \left(\frac{\mathrm{\varphi \zeta }}{2}\right)}}-\frac{b}{2a}-\frac{\varphi }{2a}\tanh \left(\frac{1}{2}\mathrm{\varphi \zeta }\right)\\ & +\frac{\mathrm{sech}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)}{\mathrm{Kcosh}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)-\frac{2a}{\varphi }\sinh \left(\frac{\mathrm{\varphi \zeta }}{2}\right)}),\end{array}$(80) where $\zeta =(\nu -ϵ)x+ϵ(\nu +1)y-\mathrm{\nu z}-\mathrm{\nu t}$ together K regarded as an integration constant. The solutions represent topological kink soliton solution and mixed dark-bright singular soliton solutions of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ). We demonstrate the streaming behaviour of solutions (79(79) $u(t,x,y,z)=A_0+A_1(-\frac{b}{2a}-\frac{\varphi }{2a}\tanh \left[\frac{1}{2}\varphi \right(\zeta +K\left)\right])$(79) ) and (80(80) $\begin{array}{rl}u(t,x,y,z)& =A_0+A_1(\phantom{\frac{\mathrm{sech}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)}{\mathrm{Kcosh}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)-\frac{2a}{\varphi }\sinh \left(\frac{\mathrm{\varphi \zeta }}{2}\right)}}-\frac{b}{2a}-\frac{\varphi }{2a}\tanh \left(\frac{1}{2}\mathrm{\varphi \zeta }\right)\\ & +\frac{\mathrm{sech}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)}{\mathrm{Kcosh}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)-\frac{2a}{\varphi }\sinh \left(\frac{\mathrm{\varphi \zeta }}{2}\right)}),\end{array}$(80) ) graphically for varying values of y with z kept constant in Figures 9, 10, 11 and 12.

3. Applications and discussion of the solutions and their graphical representations in physics and engineering

Solitons are regarded as self-enhancing wave packets that preserve their shapes while they are proliferating at a constant velocity. Solitons are naturally instigated by the cancellation of the influence of dispersive and nonlinear terms as they propagate. Presently, the investigation of solitons of nonlinear differential equations is quite active. The results secured here are solitonic in nature. For instance, elliptic solution (68(68) $\begin{array}{rl}& u(t,x,y,z)\\ & =\frac{2(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1\left)\right)}{\theta \left(\nu \right(ϵ-1)+ϵ)}\{\phantom{\frac{({\text{β}}_1-{\text{β}}_2)\mathrm{dn}\left(B\zeta |Q^2\right)E\left(\mathrm{am}\right(B\zeta |Q^2\left)|Q^2\right)}{B{Q}^2\sqrt{\mathrm{dn}(B\zeta |Q^2{)}^2}}}\frac{\{{\text{β}}_2-(1-Q^2\left){\text{β}}_1\right\}\zeta }{Q^2}\\ & +\frac{({\text{β}}_1-{\text{β}}_2)\mathrm{dn}\left(B\zeta |Q^2\right)E\left(\mathrm{am}\right(B\zeta |Q^2\left)|Q^2\right)}{B{Q}^2\sqrt{\mathrm{dn}(B\zeta |Q^2{)}^2}}\},\end{array}$(68) ) is a smooth periodic soliton solution depicted by dark-bright soliton wave profile (as can be seen in Figures 3 and 4). Moreover, the hyperbolic solution (77(77) $\begin{array}{rl}& u(t,x,y,z)\\ & =A_0+A_{1}a\left\{\frac{\cosh \left[a\right(\zeta +K\left)\right]+\sinh \left[a\right(\zeta +K\left)\right]}{\begin{array}{c}1-b\cosh \left[a\right(\zeta +K\left)\right]\\ -b\sinh \left[a\right(\zeta +K\left)\right]\end{array}}\right\}\end{array}$(77) ) is a soliton solution which produces the bright as well as dark solitonic wave structure in Figures 7 alongside Figure 8. In addition, solutions (79(79) $u(t,x,y,z)=A_0+A_1(-\frac{b}{2a}-\frac{\varphi }{2a}\tanh \left[\frac{1}{2}\varphi \right(\zeta +K\left)\right])$(79) ) and (80(80) $\begin{array}{rl}u(t,x,y,z)& =A_0+A_1(\phantom{\frac{\mathrm{sech}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)}{\mathrm{Kcosh}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)-\frac{2a}{\varphi }\sinh \left(\frac{\mathrm{\varphi \zeta }}{2}\right)}}-\frac{b}{2a}-\frac{\varphi }{2a}\tanh \left(\frac{1}{2}\mathrm{\varphi \zeta }\right)\\ & +\frac{\mathrm{sech}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)}{\mathrm{Kcosh}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)-\frac{2a}{\varphi }\sinh \left(\frac{\mathrm{\varphi \zeta }}{2}\right)}),\end{array}$(80) ) represent topological kink soliton and singular soliton of the underlying equation (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) respectively. The first two figures depict the coexistence of dark alongside bright solitons. Bright solitons occur on a zero-intensity background, whereas dark solitons emerge as an intensity submerged in an infinitely extended constant background. Furthermore, dark and bright solitons are a very important concept in science, technology and information engineering, take for instance, in the fibre ring laser mechanism, it was observed that since there is an existence of interactions between solitons, coexistence may occur when soliton pairs are propagated together in the considered medium in addition to securing dark or bright pulses, bright–bright, dark–dark or dark–bright soliton pairs in fibre lasers. Theoretically, this has been demonstrated with the use of the nonlinear Schrödinger equation [74]. The lifetime of the dark–bright soliton is revealed to be longer always than that of the dark soliton and as matter of fact becomes larger as the bright-soliton component of the dark one becomes stronger. In addition, dark–bright soliton pairs have also been observed in Kerr-type nonlinear mediums as well as optical fibre systems. Besides, in optical communication systems, dark solitons are found to be less influenced by the perturbation process, which implies that dark solitons could be preferable to bright ones. Besides, the anti-dark solitons possess profiles similar to those of the bright ones but with the exception that they exist on a nonzero background like the dark ones [75].

The anti-kink wave profile depiction of hyperbolic function (15(15) $\begin{array}{rl}& u(t,x,y,z)\\ & =z+{\displaystyle \frac{4\alpha A_2}{\theta }}\tanh (A_{2}x-{\displaystyle \frac{\theta }{4\alpha A_2}}y+A_1)+A_3\end{array}$(15) ) is exhibited in Figure 1 with the dissimilar parametric values $\theta =10$, $\alpha =4$, $A_1=0$, $A_2=1$, $A_3=5$ with variables $t=0$, $z=2$, $-0.2\le x\le 4$ as well as $-3\le y\le 5$. In the same vein, kink solution (25(25) $\begin{array}{rl}& u(t,x,y,z)\\ & ={\displaystyle \frac{4\text{β}{C}_0}{\theta }}\tanh ({\displaystyle \frac{4\text{β}{C}_0^2{C}_1}{\gamma }}t+C_{0}x+C_{2}z+C_3)+C_4,\end{array}$(25) ) is revealed in Figure 2 with unalike parameters $\theta =10$, $\alpha =4$, $\gamma =4$, $C_0=1$, $C_1=2$, $C_2=0$, $C_3=1$, $C_4=0$ where variables $y=0$, $z=1$ with $-3\le t,x\le 3$.

Figure 1. Solitary wave profile of kink solution (15(15) $\begin{array}{rl}& u(t,x,y,z)\\ & =z+{\displaystyle \frac{4\alpha A_2}{\theta }}\tanh (A_{2}x-{\displaystyle \frac{\theta }{4\alpha A_2}}y+A_1)+A_3\end{array}$(15) ) at t = 0, z = 2.

Figure 2. Solitary wave profile of kink solution (25(25) $\begin{array}{rl}& u(t,x,y,z)\\ & ={\displaystyle \frac{4\text{β}{C}_0}{\theta }}\tanh ({\displaystyle \frac{4\text{β}{C}_0^2{C}_1}{\gamma }}t+C_{0}x+C_{2}z+C_3)+C_4,\end{array}$(25) ) at y = 0, z = 1.

Figure 3. Solitary wave profile of elliptic function solution (68(68) $\begin{array}{rl}& u(t,x,y,z)\\ & =\frac{2(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1\left)\right)}{\theta \left(\nu \right(ϵ-1)+ϵ)}\{\phantom{\frac{({\text{β}}_1-{\text{β}}_2)\mathrm{dn}\left(B\zeta |Q^2\right)E\left(\mathrm{am}\right(B\zeta |Q^2\left)|Q^2\right)}{B{Q}^2\sqrt{\mathrm{dn}(B\zeta |Q^2{)}^2}}}\frac{\{{\text{β}}_2-(1-Q^2\left){\text{β}}_1\right\}\zeta }{Q^2}\\ & +\frac{({\text{β}}_1-{\text{β}}_2)\mathrm{dn}\left(B\zeta |Q^2\right)E\left(\mathrm{am}\right(B\zeta |Q^2\left)|Q^2\right)}{B{Q}^2\sqrt{\mathrm{dn}(B\zeta |Q^2{)}^2}}\},\end{array}$(68) ) at t = 0.1, y = 0.

Figure 4. Solitary wave profile of elliptic function solution (68(68) $\begin{array}{rl}& u(t,x,y,z)\\ & =\frac{2(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1\left)\right)}{\theta \left(\nu \right(ϵ-1)+ϵ)}\{\phantom{\frac{({\text{β}}_1-{\text{β}}_2)\mathrm{dn}\left(B\zeta |Q^2\right)E\left(\mathrm{am}\right(B\zeta |Q^2\left)|Q^2\right)}{B{Q}^2\sqrt{\mathrm{dn}(B\zeta |Q^2{)}^2}}}\frac{\{{\text{β}}_2-(1-Q^2\left){\text{β}}_1\right\}\zeta }{Q^2}\\ & +\frac{({\text{β}}_1-{\text{β}}_2)\mathrm{dn}\left(B\zeta |Q^2\right)E\left(\mathrm{am}\right(B\zeta |Q^2\left)|Q^2\right)}{B{Q}^2\sqrt{\mathrm{dn}(B\zeta |Q^2{)}^2}}\},\end{array}$(68) ) at t = 25, x = 25.

Now, we consider each of the secured solutions and their pictorial depictions. Graphical representation of elliptic solution (68(68) $\begin{array}{rl}& u(t,x,y,z)\\ & =\frac{2(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1\left)\right)}{\theta \left(\nu \right(ϵ-1)+ϵ)}\{\phantom{\frac{({\text{β}}_1-{\text{β}}_2)\mathrm{dn}\left(B\zeta |Q^2\right)E\left(\mathrm{am}\right(B\zeta |Q^2\left)|Q^2\right)}{B{Q}^2\sqrt{\mathrm{dn}(B\zeta |Q^2{)}^2}}}\frac{\{{\text{β}}_2-(1-Q^2\left){\text{β}}_1\right\}\zeta }{Q^2}\\ & +\frac{({\text{β}}_1-{\text{β}}_2)\mathrm{dn}\left(B\zeta |Q^2\right)E\left(\mathrm{am}\right(B\zeta |Q^2\left)|Q^2\right)}{B{Q}^2\sqrt{\mathrm{dn}(B\zeta |Q^2{)}^2}}\},\end{array}$(68) ) is presented with 3D plot, density plot as well as 2D plot in Figure 3 to display the streaming behaviour of the said solution with the unalike values of the involved constants as $\nu =-1$, $ϵ=-2$, $\text{β}=1$, $\alpha =1$, $\theta =-2$, ${\text{β}}_1=3$, ${\text{β}}_2=2$, ${\text{β}}_3=1$, Q = 0.005. Variables x and z exist in the interval $-1\le x,z\le 1$ with t = 0.1 and y = 0. We display the dynamics of elliptic-solution (68(68) $\begin{array}{rl}& u(t,x,y,z)\\ & =\frac{2(\text{β}\nu -\mathrm{\alpha ϵ}(\nu +1\left)\right)}{\theta \left(\nu \right(ϵ-1)+ϵ)}\{\phantom{\frac{({\text{β}}_1-{\text{β}}_2)\mathrm{dn}\left(B\zeta |Q^2\right)E\left(\mathrm{am}\right(B\zeta |Q^2\left)|Q^2\right)}{B{Q}^2\sqrt{\mathrm{dn}(B\zeta |Q^2{)}^2}}}\frac{\{{\text{β}}_2-(1-Q^2\left){\text{β}}_1\right\}\zeta }{Q^2}\\ & +\frac{({\text{β}}_1-{\text{β}}_2)\mathrm{dn}\left(B\zeta |Q^2\right)E\left(\mathrm{am}\right(B\zeta |Q^2\left)|Q^2\right)}{B{Q}^2\sqrt{\mathrm{dn}(B\zeta |Q^2{)}^2}}\},\end{array}$(68) ) in Figure 4 with dark–bright soliton wave profile in 3D, density as well as 2D plots with the dissimilar values of the included parameters presented as $ϵ=0$, $\nu =-1$, $\text{β}=1$, $\alpha =1$, $\theta =1$, ${\text{β}}_1=3$, ${\text{β}}_2=2$, ${\text{β}}_3=1$, Q = 0.009. In addition, variables x and z exist in the closed interval $-1.5\le x,z\le 1.5$ with t = 25 alongside x = 25. Now, we exhibit solitary wave solution (76(76) $\begin{array}{rl}& u(t,x,y,z)\\ & =A_0-A_{1}a\left\{\frac{Q_0}{\begin{array}{c}b\{Q_0+\cosh \left[a(\zeta +Q_1)\right]\\ -\sinh \left[a(\zeta +Q_1)\right]\}\end{array}}\right\},\end{array}$(76) ) in graphical forms in the usual manner in Figure 5 where we engage parameters including a = 1, b = 1, $\alpha =0.000000000001$, $\text{β}=0.00000000000001$, $A_0=10$, $A_1=0.6$, $Q_0=200$, $Q_1=20$, $ϵ=0.5$, $\nu =1$, where variables y = 1, z = 0 with $-2\le t\le 2$ and $-1\le x\le 1$. In addition, the solution is further demonstrated in Figure 6 where a = 1, b = 1, $\alpha =0.000000000001$, $\text{β}=0.00000000000001$, $A_0=0$, $A_1=20$, $Q_0=200$, $Q_1=20$, $ϵ=0.5$, $\nu =1$, with variables y = 1, z = 0, where we have $-1\le t\le 1$ and $-1\le x\le 1$.

Figure 5. Solitary wave profile of hyperbolic function solution (76(76) $\begin{array}{rl}& u(t,x,y,z)\\ & =A_0-A_{1}a\left\{\frac{Q_0}{\begin{array}{c}b\{Q_0+\cosh \left[a(\zeta +Q_1)\right]\\ -\sinh \left[a(\zeta +Q_1)\right]\}\end{array}}\right\},\end{array}$(76) ) at t = 10 x = 10.

Figure 6. Solitary wave profile of hyperbolic function solution (76(76) $\begin{array}{rl}& u(t,x,y,z)\\ & =A_0-A_{1}a\left\{\frac{Q_0}{\begin{array}{c}b\{Q_0+\cosh \left[a(\zeta +Q_1)\right]\\ -\sinh \left[a(\zeta +Q_1)\right]\}\end{array}}\right\},\end{array}$(76) ) at t = 10, x = 10.

Figure 7. Solitary wave profile of hyperbolic function solution (77(77) $\begin{array}{rl}& u(t,x,y,z)\\ & =A_0+A_{1}a\left\{\frac{\cosh \left[a\right(\zeta +K\left)\right]+\sinh \left[a\right(\zeta +K\left)\right]}{\begin{array}{c}1-b\cosh \left[a\right(\zeta +K\left)\right]\\ -b\sinh \left[a\right(\zeta +K\left)\right]\end{array}}\right\}\end{array}$(77) ) at t = 10, x = 10.

Figure 8. Solitary wave profile of hyperbolic function solution (77(77) $\begin{array}{rl}& u(t,x,y,z)\\ & =A_0+A_{1}a\left\{\frac{\cosh \left[a\right(\zeta +K\left)\right]+\sinh \left[a\right(\zeta +K\left)\right]}{\begin{array}{c}1-b\cosh \left[a\right(\zeta +K\left)\right]\\ -b\sinh \left[a\right(\zeta +K\left)\right]\end{array}}\right\}\end{array}$(77) ) at t = −10, x = −10.

Figure 9. Solitary wave profile of kink solution (79(79) $u(t,x,y,z)=A_0+A_1(-\frac{b}{2a}-\frac{\varphi }{2a}\tanh \left[\frac{1}{2}\varphi \right(\zeta +K\left)\right])$(79) ) at y = 0, z = 0.

Figure 10. Solitary wave profile of kink solution (79(79) $u(t,x,y,z)=A_0+A_1(-\frac{b}{2a}-\frac{\varphi }{2a}\tanh \left[\frac{1}{2}\varphi \right(\zeta +K\left)\right])$(79) ) at y = 1, z = 0.

Figure 11. Solitary wave profile of hyperbolic function solution (80(80) $\begin{array}{rl}u(t,x,y,z)& =A_0+A_1(\phantom{\frac{\mathrm{sech}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)}{\mathrm{Kcosh}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)-\frac{2a}{\varphi }\sinh \left(\frac{\mathrm{\varphi \zeta }}{2}\right)}}-\frac{b}{2a}-\frac{\varphi }{2a}\tanh \left(\frac{1}{2}\mathrm{\varphi \zeta }\right)\\ & +\frac{\mathrm{sech}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)}{\mathrm{Kcosh}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)-\frac{2a}{\varphi }\sinh \left(\frac{\mathrm{\varphi \zeta }}{2}\right)}),\end{array}$(80) ) at y = z = 0.

Figure 12. Solitary wave profile of hyperbolic function solution (80(80) $\begin{array}{rl}u(t,x,y,z)& =A_0+A_1(\phantom{\frac{\mathrm{sech}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)}{\mathrm{Kcosh}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)-\frac{2a}{\varphi }\sinh \left(\frac{\mathrm{\varphi \zeta }}{2}\right)}}-\frac{b}{2a}-\frac{\varphi }{2a}\tanh \left(\frac{1}{2}\mathrm{\varphi \zeta }\right)\\ & +\frac{\mathrm{sech}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)}{\mathrm{Kcosh}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)-\frac{2a}{\varphi }\sinh \left(\frac{\mathrm{\varphi \zeta }}{2}\right)}),\end{array}$(80) ) at y = 2, z = 0.

Next, we contemplate the hyperbolic function solution (77(77) $\begin{array}{rl}& u(t,x,y,z)\\ & =A_0+A_{1}a\left\{\frac{\cosh \left[a\right(\zeta +K\left)\right]+\sinh \left[a\right(\zeta +K\left)\right]}{\begin{array}{c}1-b\cosh \left[a\right(\zeta +K\left)\right]\\ -b\sinh \left[a\right(\zeta +K\left)\right]\end{array}}\right\}\end{array}$(77) ). The streaming pattern of soliton solution (77(77) $\begin{array}{rl}& u(t,x,y,z)\\ & =A_0+A_{1}a\left\{\frac{\cosh \left[a\right(\zeta +K\left)\right]+\sinh \left[a\right(\zeta +K\left)\right]}{\begin{array}{c}1-b\cosh \left[a\right(\zeta +K\left)\right]\\ -b\sinh \left[a\right(\zeta +K\left)\right]\end{array}}\right\}\end{array}$(77) ) is displayed with dark soliton wave profile in 3D plot, density plot and 2D plots and presented in Figure 7 with diverse values of the engaged parameters including $\text{β}=1$, $ϵ=0$, $\nu =1$, $\alpha =0.000000000001$, $\theta =1$, $\gamma =4$, b = 1, K = 0, $A_0=0$. Also, variables y and z occur in the closed interval $-1.1\le y,z\le 1.1$ with t = 10 alongside x = 10. Furthermore, Figure 8 depicts the bright soliton wave profile of (77(77) $\begin{array}{rl}& u(t,x,y,z)\\ & =A_0+A_{1}a\left\{\frac{\cosh \left[a\right(\zeta +K\left)\right]+\sinh \left[a\right(\zeta +K\left)\right]}{\begin{array}{c}1-b\cosh \left[a\right(\zeta +K\left)\right]\\ -b\sinh \left[a\right(\zeta +K\left)\right]\end{array}}\right\}\end{array}$(77) ) exhibited in 3D plot, density plot as well as 2D plot. Various values of the involved parameters are given as $\nu =-10$, $\alpha =10$, $\text{β}=10$, $ϵ=0$, $\theta =1$, $\gamma =4$, b = 1, K = 0, $A_0=0$. Besides, variables y and z eventuate in the closed interval $-1\le y,z\le 1$ with t = −10 and x = −10. We can easily add here that Figures 7 and 8 represent a one single-peak soliton wave of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ). It is worthy of note to include here that the association of Figures 7 and 8 to the hyperbolic solution (77(77) $\begin{array}{rl}& u(t,x,y,z)\\ & =A_0+A_{1}a\left\{\frac{\cosh \left[a\right(\zeta +K\left)\right]+\sinh \left[a\right(\zeta +K\left)\right]}{\begin{array}{c}1-b\cosh \left[a\right(\zeta +K\left)\right]\\ -b\sinh \left[a\right(\zeta +K\left)\right]\end{array}}\right\}\end{array}$(77) ) is in consonance with already established fact because it is well known that hyperbolic secant characterizes bright soliton profile. This bright soliton solution usually takes a structure of bell-shaped figure which propagates undistorted without any alteration or altercation in shape for arbitrarily long distances. Nonetheless, dark soliton solutions which are invariably configured also as topological optical solitons, are represented by hyperbolic tangent. We represent the dynamics of topological kink soliton solution (79(79) $u(t,x,y,z)=A_0+A_1(-\frac{b}{2a}-\frac{\varphi }{2a}\tanh \left[\frac{1}{2}\varphi \right(\zeta +K\left)\right])$(79) ) in Figure 9. Actually, a kink ascends from one asymptotic state to another. The kink wave is a paramount aspect of numerous physical phenomena that consist of impulsive systems, self-reinforcing, as well as reaction–diffusion–advection. It is very important to explore kink wave in relation to solutions of partial differential equations. The 3D plot, density plot and 2D plot of (79(79) $u(t,x,y,z)=A_0+A_1(-\frac{b}{2a}-\frac{\varphi }{2a}\tanh \left[\frac{1}{2}\varphi \right(\zeta +K\left)\right])$(79) ) are given in Figure 9 with unalike values of the various included parameters stated as $\alpha =0.000000000001$, $\text{β}=1$, $\nu =2.5$, $ϵ=0$, $\theta =-0.06667$, $\gamma =6.25$, b = 2, $\varphi =2$, K = −0.9, d = 1, $A_0=100$. Also, variables t as well as x eventuate in the closed interval $0\le t\le 1$ and $0\le x\le 2$ with y = 0 and z = 0. Moreover, we give the 3D, density and 2D plots of soliton (79(79) $u(t,x,y,z)=A_0+A_1(-\frac{b}{2a}-\frac{\varphi }{2a}\tanh \left[\frac{1}{2}\varphi \right(\zeta +K\left)\right])$(79) ) in Figure 10 with dissimilar values of the engaged parameters as $\alpha =0.000000000001$, $\nu =-0.8$, $\text{β}=1$, $ϵ=0$, $\theta =-0.02$, $\gamma =6.25$, b = 2, $\varphi =2$, K = −0.9, d = 1, $A_0=100$. Besides, variables t as well as x existing in the closed interval $0\le t\le 1$ and $0\le x\le 2$ with y = 1 and z = 0.

The streaming behaviour of singular soliton solution (80(80) $\begin{array}{rl}u(t,x,y,z)& =A_0+A_1(\phantom{\frac{\mathrm{sech}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)}{\mathrm{Kcosh}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)-\frac{2a}{\varphi }\sinh \left(\frac{\mathrm{\varphi \zeta }}{2}\right)}}-\frac{b}{2a}-\frac{\varphi }{2a}\tanh \left(\frac{1}{2}\mathrm{\varphi \zeta }\right)\\ & +\frac{\mathrm{sech}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)}{\mathrm{Kcosh}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)-\frac{2a}{\varphi }\sinh \left(\frac{\mathrm{\varphi \zeta }}{2}\right)}),\end{array}$(80) ) is presented in Figures 11 and 12. Singular solitons can be associated with a solitary wave whose centre is an imaginary position. In addition, their intensity turn out to be stronger and as a result they are unstable. The 3D plot, density plot as well as 2D plot of singular soliton (80(80) $\begin{array}{rl}u(t,x,y,z)& =A_0+A_1(\phantom{\frac{\mathrm{sech}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)}{\mathrm{Kcosh}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)-\frac{2a}{\varphi }\sinh \left(\frac{\mathrm{\varphi \zeta }}{2}\right)}}-\frac{b}{2a}-\frac{\varphi }{2a}\tanh \left(\frac{1}{2}\mathrm{\varphi \zeta }\right)\\ & +\frac{\mathrm{sech}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)}{\mathrm{Kcosh}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)-\frac{2a}{\varphi }\sinh \left(\frac{\mathrm{\varphi \zeta }}{2}\right)}),\end{array}$(80) ) is projected in Figure 11. The parameters invoked take up the values as $\nu =-2$, $\alpha =0.000000000001$, $\text{β}=1$, $ϵ=0$, $\theta =-0.06667$, $\gamma =6.25$, b = 2, $\varphi =2$, K = −0.9, d = 1, $A_0=100$. Also, variables t as well as x eventuate in closed interval $0\le t\le 1$ and $0\le x\le 2$ with y = 0 and z = 0. We observe that the points of singularities for the solution in association with those values exist in the spatial domain $1\le x\le 2$. Lastly, the dynamics of soliton (80(80) $\begin{array}{rl}u(t,x,y,z)& =A_0+A_1(\phantom{\frac{\mathrm{sech}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)}{\mathrm{Kcosh}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)-\frac{2a}{\varphi }\sinh \left(\frac{\mathrm{\varphi \zeta }}{2}\right)}}-\frac{b}{2a}-\frac{\varphi }{2a}\tanh \left(\frac{1}{2}\mathrm{\varphi \zeta }\right)\\ & +\frac{\mathrm{sech}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)}{\mathrm{Kcosh}\left(\frac{\mathrm{\varphi \zeta }}{2}\right)-\frac{2a}{\varphi }\sinh \left(\frac{\mathrm{\varphi \zeta }}{2}\right)}),\end{array}$(80) ) is further exhibited in Figure 12 via 3D plot, density plot and 2D plot representations. Varying values of the involved parameters are stated as $\alpha =0.000000000001$, $\nu =-0.8$, $\text{β}=1$, $ϵ=0$, $\theta =-0.02$, $\gamma =6.25$, b = 2, $\varphi =2$, K = 0.009, d = 1, $A_0=100$. More so, variables t as well as x occur in the closed-interval $0\le t\le 1$ and $0\le x\le 2$ with y = 2 and z = 0. The singularity in this case exists at the spatial point x = 1.5. We note that Figures 5 and 6 produce multi-peak soliton waves.

Next, we endeavour to examine the connectivity ensuing between our graphic depictions and real-world problems. To have a good understanding of complex situations and then obtain real-world practical applications, mathematical visualization is needed. We particularly note that kink-shaped structures (as can be seen in Figures 1, 2, 9 and 10) are prominent in the research work, which in the real world perfectly depicts the shape of waterfalls (see the images in Figures 13 and 14).

Figure 13. Connection of the various kink-shaped structures to the real-world application through waterfalls. https://www.newyorkupstate.com/outdoors/2020/10/17-must-see-waterfalls-that-should-be-on-your-upstate-ny-bucket-list.html.

Figure 14. Relating the various kink-shaped structures to the real-world application through waterfalls. https://www.americanrivers.org/2022/08/what-makes-a-waterfall-a-waterfall/.

By visualizing the structure of kink-wave viz-a-viz its symbolic implication in waterfalls, it could be of help to ocean scientists and engineers in studying the dynamics of fluids. Besides, kink solutions have a range of uses including resolving issues and bugs, in code as well as tackling bottlenecks or unexpected behaviours, in technology and software systems. In the realm of business management, they assist managers in overcoming challenges and finding solutions to market demands and competition. Additionally, they aid healthcare professionals in handling complications during treatments. The applications of kink solutions are extensive spanning across aspects of life. They involve adaptability, problem-solving skills and the ability to overcome obstacles thus contributing to progress and advancement in fields.

3.1. Significance and application of exact solutions in science and engineering fields

This work presents various dynamics of exact solutions to the underlying problem (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) which range from hyperbolic, logarithmic, algebraic (with and without arbitrary functions) to elliptic function solutions. Therefore, it is expedient to present the significant of exact solutions in science and engineering fields. In general relativity, an exact solution is commonly referred to as a solution of the Einstein field equations in which its derivation does not include the engagement of simplifying assumptions. However, the derivation starting point may be an idealized situation, just for instance, the case of a perfectly spherical shape of matter. In mathematics, securing an exact solution amounts to achieving a Lorentzian manifold that has been equipped with tensor fields modelling states of ordinary matter. These could appear as a fluid or non-gravitational fields (classical) such as the electromagnetic field [76].

Exact (closed-form) solutions associated to mathematical models play an exerting crucial role in having a proper perspective (understanding) of qualitative features related to many processes and phenomena arising in various areas of natural science. Besides, exact solutions of nonlinear DEs graphically exhibit and allow unravelling the mechanisms which are included in various complex nonlinear phenomena such as multiplicity or absence of steady states under diverse conditions, spatial localization of transfer processes, existence of peaking regimes, to mention but a few. Moreover, simple solutions are often involved in teaching many courses. This enables the use of a good number of specific examples depicting basic tenets of a theory that admit mathematical formulation (such as can be found in hydrodynamics, heat and mass transfer theory, gas dynamics, nonlinear optics, wave theory and various other fields of research).

Furthermore, even those seemingly special exact solutions with no clear physical meaning can be utilized as test problems for the verification of the consistency as well as estimation of errors of various asymptotic, numerical, as well as approximate analytical techniques. Significantly, exact solutions can also be employed as a basis for the perfection and testing of computer algebra software packages in attaining solution to DEs. These packages include Mathematica, MATLAB, Maple, CONVODE and diverse others.

It is pertinent to say that many models found in biology, physics and chemistry comprise of empirical functions or empirical parameters. In addition, exact solutions give researchers the opportunity of designing and also running experiments, through the creation of appropriate natural (boundary and initial) conditions, in determining the aforementioned functions or parameters [77].

Remark 3.1

We note that an exact solution is typically constructed through the use of mathematical equations or proofs. Besides, it is commonly represented through a symbolic representation or closed-form expression. A solution is exact in the sense that for any value of the variables in a given problem, it is valid, and it can also be evaluated with any precision. Therefore, an exact solution refers to a symbolic representation of a value that precisely solves a given equation. However, an equation is said to be a closed form if it solves a given problem regarding mathematical operations and functions from a given generally accepted set. Take, for instance, an infinite sum that would generally not be considered closed-form. Thus a solution to a given problem that can be presented in “closed form” with regard to known functions, constants and so on is usually called an analytic solution.

4. Conservation laws of 3D-genBSMe (eqn5)

This section gives the result gained when the conserved quantities of 3D-genBSMe (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) were derived via the utilization of the long-established Noether's theorem [78] as highlighted in [21].

4.1. Establishment of conserved quantities via Noether's theorem

On utilizing the long-established Noether's theorem [78] to calculate the conserved vectors of the 3D-genBSMe (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ), the Noether's theorem is stated in the first place, to have some background ideas about the technique.

Theorem 4.1

If Q is regarded as a Noether symmetry operator corresponding to Lagrangian $\mathcal{L}$ in such a way that $\mathcal{L}=\mathcal{L}(x,\mathrm{\Psi },{\mathrm{\Psi }}_{\left(1\right)},\dots ,{\mathrm{\Psi }}_{(k-1)})$ of mth-order system of k PDEs expressed as (81) ${\Omega }^{\alpha }=(x,\mathrm{\Psi },{\mathrm{\Psi }}_{\left(1\right)},\dots ,{\mathrm{\Psi }}_{\left(m\right)})=0,\mathrm{with}\alpha =1,\dots ,k,$(81) with x as well as Ψ representing independent together with dependent variables, respectively, then there exists a vector called $C=(C^1,C^2,\dots ,C^m)$ relating to $\mathcal{L}$ such that (82) $C^i=N^i\mathcal{L}-B^i,i=1,\dots ,m,$(82) with gauge function $B^i$ dependent on x, is accented to as conserved vector for (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) related to operator Q [78].

The Noether operators are given by (83) $\begin{array}{rl}{N}^i& ={\xi }^i+W^{\alpha }\frac{\delta }{\delta {\mathrm{\Psi }}_i^{\alpha }}+\sum _{s\ge 1}{D}_{i_1}\dots D_{i_s}\left(W^{\alpha }\right)\frac{\delta }{\delta {\mathrm{\Psi }}_{i{i}_1{i}_2\dots i_s}^{\alpha }},\\ & i=1,\dots ,m,\end{array}$(83) and Euler–Lagrange  operators are defined as [21] (84) $\begin{array}{rl}\frac{\delta }{\delta {\mathrm{\Psi }}^{\alpha }}& =\frac{\mathrm{\partial }}{\mathrm{\partial }{\mathrm{\Psi }}^{\alpha }}+\sum _{s\ge 1}(-1{)}^s{D}_{i_1}\dots D_{i_s}\frac{\mathrm{\partial }}{\mathrm{\partial }{\mathrm{\Psi }}_{i_1{i}_2\dots i_s}^{\alpha }},\\ & \alpha =1,\dots ,k.\end{array}$(84)

Lemma 4.1

The 3D-genBSMe (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) forms the Euler–Lagrange relation with functional (85) $\begin{array}{rl}J\left(u\right)& ={\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}\mathcal{L}(t,x,y,z,u_t,u_x,u_y,u_z,\\ & u_{\mathrm{xx}},u_{\mathrm{xy}},u_{\mathrm{xz}})\mathrm{d}t\mathrm{d}x\mathrm{d}y\mathrm{d}z,\end{array}$(85) where Lagrange-conforming function of $\mathcal{L}$ is expressed as (86) $\begin{array}{rl}\mathcal{L}& =\frac{1}{2}\gamma u_t{u}_x+\frac{1}{2}\alpha u_{\mathrm{xx}}{u}_{\mathrm{xy}}-\frac{1}{2}\theta u_x^2{u}_y\\ & +\frac{1}{2}\text{β}{u}_{\mathrm{xx}}{u}_{\mathrm{xz}}-\frac{1}{2}\theta u_x^2{u}_z.\end{array}$(86)

Proof.

We comment that second-order Lagrangian of the understudy model is explicated via (86(86) $\begin{array}{rl}\mathcal{L}& =\frac{1}{2}\gamma u_t{u}_x+\frac{1}{2}\alpha u_{\mathrm{xx}}{u}_{\mathrm{xy}}-\frac{1}{2}\theta u_x^2{u}_y\\ & +\frac{1}{2}\text{β}{u}_{\mathrm{xx}}{u}_{\mathrm{xz}}-\frac{1}{2}\theta u_x^2{u}_z.\end{array}$(86) ), due to the fact that Euler–Lagrange relation $\delta \mathcal{L}/\mathrm{\delta u}=0$ is satisfied as anticipated.

Moreover, we note that in Section 2, the 3D-genBSMe (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) has 11 Lie point symmetries which are expressed in (11(11) $\begin{array}{rl}{\mathcal{P}}_1& =\frac{\mathrm{\partial }}{\mathrm{\partial }t},{\mathcal{P}}_2=\frac{\mathrm{\partial }}{\mathrm{\partial }x},{\mathcal{P}}_3=\frac{\mathrm{\partial }}{\mathrm{\partial }y},\\ {\mathcal{P}}_4& =\frac{\mathrm{\partial }}{\mathrm{\partial }z},{\mathcal{P}}_5=G_1\left(t\right)\frac{\mathrm{\partial }}{\mathrm{\partial }u},{\mathcal{P}}_6=G(t,z-y)\frac{\mathrm{\partial }}{\mathrm{\partial }u},\\ {\mathcal{P}}_7& =u\frac{\mathrm{\partial }}{\mathrm{\partial }u}-2t\frac{\mathrm{\partial }}{\mathrm{\partial }t}-x\frac{\mathrm{\partial }}{\mathrm{\partial }x},{\mathcal{P}}_8=t\frac{\mathrm{\partial }}{\mathrm{\partial }t}+y\frac{\mathrm{\partial }}{\mathrm{\partial }y}+z\frac{\mathrm{\partial }}{\mathrm{\partial }z},\\ {\mathcal{P}}_9& =x\frac{\mathrm{\partial }}{\mathrm{\partial }u}-\frac{2\theta }{\gamma }t\frac{\mathrm{\partial }}{\mathrm{\partial }y}-\frac{2\theta }{\gamma }t\frac{\mathrm{\partial }}{\mathrm{\partial }z}\\ {\mathcal{P}}_{10}& =(y+z)G_2^{\mathrm{\prime }}\left(t\right)\frac{\mathrm{\partial }}{\mathrm{\partial }u}-\frac{2\theta }{\gamma }{G}_2\left(t\right)\frac{\mathrm{\partial }}{\mathrm{\partial }x},\\ {\mathcal{P}}_{11}& =\left(\frac{\alpha -2\text{β}}{\text{β}}\right)t\frac{\mathrm{\partial }}{\mathrm{\partial }t}+\left(\frac{\alpha -\text{β}}{\text{β}}\right)x\frac{\mathrm{\partial }}{\mathrm{\partial }x}\\ & -\left(\frac{\alpha (y-z)+\text{β}y}{\text{β}}\right)\frac{\mathrm{\partial }}{\mathrm{\partial }y}-y\frac{\mathrm{\partial }}{\mathrm{\partial }z}.\end{array}$(11) ). By utilizing the result given in [21], we conclude that vector fields ${\mathcal{P}}_1,{\mathcal{P}}_2,{\mathcal{P}}_3,{\mathcal{P}}_4,{\mathcal{P}}_5,{\mathcal{P}}_6,{\mathcal{P}}_7,{\mathcal{P}}_8,{\mathcal{P}}_9,{\mathcal{P}}_{10}$ and ${\mathcal{P}}_{11}$ (where the arbitrary functions $G_1\left(t\right)$, $G(t,z-y)$ and $G_2\left(t\right)$ satisfy the derivatives $G_1^{\mathrm{\prime }}\left(t\right)=0$, $G_t(t,z-y)=0$ and $G_2^{\mathrm{\prime \prime }}\left(t\right)=0$ respectively) are variational symmetries. Therefore, the conserved vectors $(C^t,C^x,C^y,C^z)$ related to 11 Noether symmetries ${\mathcal{P}}_1,{\mathcal{P}}_2,{\mathcal{P}}_3,{\mathcal{P}}_4,{\mathcal{P}}_5,$ ${\mathcal{P}}_6,{\mathcal{P}}_7,{\mathcal{P}}_8,{\mathcal{P}}_9,{\mathcal{P}}_{10}$ and ${\mathcal{P}}_{11}$ with the aid of [79] $\begin{array}{rl}{C}^k& =\mathcal{L}{\tau }^k+({\xi }^{\alpha }-{\psi }_{x^j}^{\alpha }{\tau }^j)(\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{\psi }_{x^k}^{\alpha }}-\sum _{l=1}^k{D}_{x^l}\left(\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{\psi }_{x^l{x}^k}^{\alpha }}\right))\\ & +\sum _{l=k}^n({\eta }_l^{\alpha }-{\psi }_{x^l{x}^j}^{\alpha }{\tau }^j)\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{\psi }_{x^k{x}^l}^{\alpha }}\end{array}$ are, respectively, given by $\begin{array}{rl}{C}_1^t& =\frac{1}{2}\alpha u_{\mathrm{xx}}{u}_{\mathrm{xy}}-\frac{1}{2}\theta u_x^2{u}_y+\frac{1}{2}\text{β}{u}_{\mathrm{xx}}{u}_{\mathrm{xz}}-\frac{1}{2}\theta u_x^2{u}_z,\\ C_1^x& =\frac{3}{4}\alpha u_t{u}_{\mathrm{xxy}}-\frac{1}{2}\gamma u_t^2-\frac{1}{4}\alpha u_{\mathrm{xx}}{u}_{\mathrm{ty}}-\frac{1}{2}\alpha u_{\mathrm{tx}}{u}_{\mathrm{xy}}\\ & +\theta u_t{u}_x{u}_y+\frac{3}{4}\text{β}{u}_t{u}_{\mathrm{xxz}}\\ & -\frac{1}{4}\text{β}{u}_{\mathrm{xx}}{u}_{\mathrm{tz}}-\frac{1}{2}\text{β}{u}_{\mathrm{tx}}{u}_{\mathrm{xz}}+\theta u_t{u}_x{u}_z,\\ C_1^y& =\frac{1}{4}\alpha u_t{u}_{\mathrm{xxx}}-\frac{1}{4}\alpha u_{\mathrm{xx}}{u}_{\mathrm{tx}}+\frac{1}{2}\theta u_t{u}_x^2,\\ C_1^z& =\frac{1}{4}\text{β}{u}_t{u}_{\mathrm{xxx}}-\frac{1}{4}\text{β}{u}_{\mathrm{xx}}{u}_{\mathrm{tx}}+\frac{1}{2}\theta u_t{u}_x^2;\\ C_2^t& =-\frac{1}{2}\gamma u_x^2,\\ C_2^x& =\frac{3}{4}\alpha u_x{u}_{\mathrm{xxy}}-\frac{1}{4}\alpha u_{\mathrm{xx}}{u}_{\mathrm{xy}}+\frac{1}{2}\theta u_x^2{u}_y+\frac{3}{4}\text{β}{u}_x{u}_{\mathrm{xxz}}\\ & -\frac{1}{4}\text{β}{u}_{\mathrm{xx}}{u}_{\mathrm{xz}}+\frac{1}{2}\theta u_x^2{u}_z,\\ C_2^y& =\frac{1}{4}\alpha u_{\mathrm{xxx}}{u}_x-\frac{1}{4}\alpha u_{\mathrm{xx}}^2+\frac{1}{2}\theta u_x^3,\\ C_2^z& =\frac{1}{4}\text{β}{u}_{\mathrm{xxx}}{u}_x-\frac{1}{4}\text{β}{u}_{\mathrm{xx}}^2+\frac{1}{2}\theta u_x^3;\\ C_3^t& =-\frac{1}{2}\gamma u_x{u}_y,\end{array}$ $\begin{array}{rl}{C}_3^x& =\frac{3}{4}\alpha u_y{u}_{\mathrm{xxy}}-\frac{1}{2}\gamma u_t{u}_y-\frac{1}{2}\alpha u_{\mathrm{xy}}^2-\frac{1}{4}\alpha u_{\mathrm{xx}}{u}_{\mathrm{yy}}\\ & +\theta u_x{u}_y^2+\frac{3}{4}\text{β}{u}_y{u}_{\mathrm{xxz}}\\ & -\frac{1}{2}\text{β}{u}_{\mathrm{xy}}{u}_{\mathrm{xz}}-\frac{1}{4}\text{β}{u}_{\mathrm{xx}}{u}_{\mathrm{yz}}+\theta u_x{u}_y{u}_z,\\ C_3^y& =\frac{1}{2}\gamma u_t{u}_x+\frac{1}{4}\alpha u_{\mathrm{xx}}{u}_{\mathrm{xy}}+\frac{1}{4}\alpha u_{\mathrm{xxx}}{u}_y\\ & +\frac{1}{2}\text{β}{u}_{\mathrm{xx}}{u}_{\mathrm{xz}}-\frac{1}{2}\theta u_x^2{u}_z,\\ C_3^z& =\frac{1}{4}\text{β}{u}_{\mathrm{xxx}}{u}_y-\frac{1}{4}\text{β}{u}_{\mathrm{xx}}{u}_{\mathrm{xy}}+\frac{1}{2}\theta u_x^2{u}_y;\\ C_4^t& =-\frac{1}{2}\gamma u_x{u}_z,\\ C_4^x& =\frac{3}{4}\alpha u_z{u}_{\mathrm{xxy}}-\frac{1}{2}\gamma u_t{u}_z-\frac{1}{2}\alpha u_{\mathrm{xy}}{u}_{\mathrm{xz}}\\ & -\frac{1}{4}\alpha u_{\mathrm{xx}}{u}_{\mathrm{yz}}+\theta u_x{u}_y{u}_z\\ & +\frac{3}{4}\text{β}{u}_z{u}_{\mathrm{xxz}}-\frac{1}{2}\text{β}{u}_{\mathrm{xz}}^2-\frac{1}{4}\text{β}{u}_{\mathrm{xx}}{u}_{\mathrm{zz}}+\theta u_x{u}_z^2,\\ C_4^y& =-\frac{1}{4}\alpha u_{\mathrm{xx}}{u}_{\mathrm{xz}}+\frac{1}{4}\alpha u_{\mathrm{xxx}}{u}_z+\frac{1}{2}\theta u_x^2{u}_z,\end{array}$ $\begin{array}{rl}{C}_4^z& =\frac{1}{2}\gamma u_t{u}_x+\frac{1}{2}\alpha u_{\mathrm{xx}}{u}_{\mathrm{xy}}-\frac{1}{2}\theta u_x^2{u}_y\\ & +\frac{1}{4}\text{β}{u}_{\mathrm{xx}}{u}_{\mathrm{xz}}+\frac{1}{4}\text{β}{u}_{\mathrm{xxx}}{u}_z;\\ C_5^t& =\frac{1}{2}\gamma G_1\left(t\right)u_x,\\ C_5^x& =\frac{1}{2}\gamma G_1\left(t\right)u_t-\frac{3}{4}\alpha G_1\left(t\right)u_{\mathrm{xxy}}\\ & -\theta G_1\left(t\right)u_x{u}_y-\frac{3}{4}\text{β}{G}_1\left(t\right)u_{\mathrm{xxz}}\\ & -\theta G_1\left(t\right)u_x{u}_z-\frac{1}{2}\gamma G_1^{\mathrm{\prime }}\left(t\right)u,\\ C_5^y& =-\frac{1}{4}\alpha G_1\left(t\right)u_{\mathrm{xxx}}-\frac{1}{2}\theta G_1\left(t\right)u_x^2,\\ C_5^z& =-\frac{1}{4}\text{β}{G}_1\left(t\right)u_{\mathrm{xxx}}-\frac{1}{2}\theta G_1\left(t\right)u_x^2;\\ C_6^t& =\frac{1}{2}\gamma u_{x}G(t,z-y),\\ C_6^x& =-\frac{3}{4}\alpha u_{\mathrm{xxy}}G(t,z-y)-\frac{3}{4}\text{β}{u}_{\mathrm{xxz}}G(t,z-y)\\ & -\theta u_x{u}_{z}G(t,z-y)-\theta u_x{u}_{y}G(t,z-y)\\ & +\frac{1}{2}\gamma u_{t}G(t,z-y)-\frac{1}{4}\alpha G_y(t,z-y)u_{\mathrm{xx}}\\ & +\frac{1}{4}\text{β}{G}_z(t,z-y)u_{\mathrm{xx}}-\frac{1}{2}\gamma G_t(t,z-y)u,\end{array}$ $\begin{array}{rl}{C}_6^y& =-\frac{1}{4}\alpha u_{\mathrm{xxx}}G(t,z-y)-\frac{1}{2}\theta u_x^{2}G(t,z-y),\\ C_6^z& =-\frac{1}{4}\text{β}{u}_{\mathrm{xxx}}G(t,z-y)-\frac{1}{2}\theta u_x^{2}G(t,z-y);\\ C_7^t& =\mathrm{\alpha t}{u}_{\mathrm{xx}}{u}_{\mathrm{xy}}-\mathrm{\theta t}{u}_x^2{u}_y+\text{β}t{u}_{\mathrm{xx}}{u}_{\mathrm{xz}}\\ & -\mathrm{\theta t}{u}_x^2{u}_z-\frac{1}{2}\mathrm{\gamma u}{u}_x-\frac{1}{2}\mathrm{\gamma x}{u}_x^2,\\ C_7^x& =\frac{3}{4}\mathrm{\alpha u}{u}_{\mathrm{xxy}}+\frac{3}{4}\text{β}u{u}_{\mathrm{xxz}}-\frac{1}{2}\mathrm{\gamma u}{u}_t+\mathrm{\theta u}{u}_x{u}_z\\ & +\mathrm{\theta u}{u}_x{u}_y-\mathrm{\gamma t}{u}_t^2+\frac{3}{2}\mathrm{\alpha t}{u}_t{u}_{\mathrm{xxy}}-\frac{1}{2}\mathrm{\alpha t}{u}_{\mathrm{xx}}{u}_{\mathrm{ty}}\\ & -\mathrm{\alpha t}{u}_{\mathrm{tx}}{u}_{\mathrm{xy}}+2\mathrm{\theta t}{u}_t{u}_x{u}_y+\frac{3}{2}\text{β}t{u}_t{u}_{\mathrm{xxz}}\\ & -\frac{1}{2}\text{β}t{u}_{\mathrm{xx}}{u}_{\mathrm{tz}}-\text{β}t{u}_{\mathrm{tx}}{u}_{\mathrm{xz}}+2\mathrm{\theta t}{u}_t{u}_x{u}_z-\alpha u_x{u}_{\mathrm{xy}}\\ & +\frac{3}{4}\mathrm{\alpha x}{u}_x{u}_{\mathrm{xxy}}-\frac{1}{4}\alpha u_{\mathrm{xx}}{u}_y-\frac{1}{4}\mathrm{\alpha x}{u}_{\mathrm{xx}}{u}_{\mathrm{xy}}\\ & +\frac{1}{2}\mathrm{\theta x}{u}_x^2{u}_y-\text{β}{u}_x{u}_{\mathrm{xz}}+\frac{3}{4}\text{β}x{u}_x{u}_{\mathrm{xxz}}\\ & -\frac{1}{4}\text{β}{u}_{\mathrm{xx}}{u}_z-\frac{1}{4}\text{β}x{u}_{\mathrm{xx}}{u}_{\mathrm{xz}}+\frac{1}{2}\mathrm{\theta x}{u}_x^2{u}_z,\end{array}$ $\begin{array}{rl}{C}_7^y& =\frac{1}{4}\mathrm{\alpha u}{u}_{\mathrm{xxx}}+\frac{1}{2}\mathrm{\theta u}{u}_x^2+\frac{1}{2}\mathrm{\alpha t}{u}_t{u}_{\mathrm{xxx}}\\ & -\frac{1}{2}\mathrm{\alpha t}{u}_{\mathrm{xx}}{u}_{\mathrm{tx}}+\mathrm{\theta t}{u}_t{u}_x^2-\frac{1}{2}\alpha u_{\mathrm{xx}}{u}_x\\ & +\frac{1}{4}\mathrm{\alpha x}{u}_{\mathrm{xxx}}{u}_x-\frac{1}{4}\mathrm{\alpha x}{u}_{\mathrm{xx}}^2+\frac{1}{2}\mathrm{\theta x}{u}_x^3,\\ C_7^z& =\frac{1}{4}\text{β}u{u}_{\mathrm{xxx}}+\frac{1}{2}\mathrm{\theta u}{u}_x^2+\frac{1}{2}\text{β}t{u}_t{u}_{\mathrm{xxx}}\\ & -\frac{1}{2}\text{β}t{u}_{\mathrm{xx}}{u}_{\mathrm{tx}}+\mathrm{\theta t}{u}_t{u}_x^2-\frac{1}{2}\text{β}{u}_{\mathrm{xx}}{u}_x\\ & +\frac{1}{4}\text{β}x{u}_{\mathrm{xxx}}{u}_x-\frac{1}{4}\text{β}x{u}_{\mathrm{xx}}^2+\frac{1}{2}\mathrm{\theta x}{u}_x^3;\\ C_8^t& =\frac{1}{2}\mathrm{\alpha t}{u}_{\mathrm{xx}}{u}_{\mathrm{xy}}-\frac{1}{2}\mathrm{\theta t}{u}_x^2{u}_y+\frac{1}{2}\text{β}t{u}_{\mathrm{xx}}{u}_{\mathrm{xz}}\\ & -\frac{1}{2}\mathrm{\theta t}{u}_x^2{u}_{,z}-\frac{1}{2}\mathrm{\gamma y}{u}_x{u}_y-\frac{1}{2}\mathrm{\gamma z}{u}_x{u}_z,\end{array}$ $\begin{array}{rl}{C}_8^x& =\frac{3}{4}\mathrm{\alpha t}{u}_t{u}_{\mathrm{xxy}}-\frac{1}{2}\mathrm{\gamma t}{u}_t^2-\frac{1}{4}\mathrm{\alpha t}{u}_{\mathrm{xx}}{u}_{\mathrm{ty}}\\ & -\frac{1}{2}\mathrm{\alpha t}{u}_{\mathrm{tx}}{u}_{\mathrm{xy}}+\mathrm{\theta t}{u}_t{u}_x{u}_y+\frac{3}{4}\text{β}t{u}_t{u}_{\mathrm{xxz}}\\ & -\frac{1}{4}\text{β}t{u}_{\mathrm{xx}}{u}_{\mathrm{tz}}-\frac{1}{2}\text{β}t{u}_{\mathrm{tx}}{u}_{\mathrm{xz}}+\mathrm{\theta t}{u}_t{u}_x{u}_z-\frac{1}{2}\mathrm{\gamma y}{u}_t{u}_y\\ & -\frac{1}{2}\mathrm{\gamma z}{u}_t{u}_z-\frac{1}{2}\mathrm{\alpha y}{u}_{\mathrm{xy}}^2\\ & -\frac{1}{4}\alpha u_{\mathrm{xx}}{u}_y-\frac{1}{4}\mathrm{\alpha y}{u}_{\mathrm{xx}}{u}_{\mathrm{yy}}+\frac{3}{4}\mathrm{\alpha y}{u}_y{u}_{\mathrm{xxy}}\\ & +\mathrm{\theta y}{u}_x{u}_y^2+\frac{3}{4}\mathrm{\alpha z}{u}_z{u}_{\mathrm{xxy}}\\ & -\frac{1}{2}\mathrm{\alpha z}{u}_{\mathrm{xy}}{u}_{\mathrm{xz}}-\frac{1}{4}\mathrm{\alpha z}{u}_{\mathrm{xx}}{u}_{\mathrm{yz}}-\frac{1}{2}\text{β}y{u}_{\mathrm{xy}}{u}_{\mathrm{xz}}\\ & -\frac{1}{4}\text{β}y{u}_{\mathrm{xx}}{u}_{\mathrm{yz}}+\frac{3}{4}\text{β}y{u}_y{u}_{\mathrm{xxz}}\\ & +\mathrm{\theta y}{u}_x{u}_y{u}_z+\mathrm{\theta z}{u}_x{u}_y{u}_z-\frac{1}{4}\text{β}{u}_{\mathrm{xx}}{u}_z\\ & +\frac{3}{4}\text{β}z{u}_z{u}_{\mathrm{xxz}}-\frac{1}{2}\text{β}z{u}_{\mathrm{xz}}^2-\frac{1}{4}\text{β}z{u}_{\mathrm{xx}}{u}_{\mathrm{zz}}+\mathrm{\theta z}{u}_x{u}_z^2,\\ C_8^y& =\frac{1}{4}\mathrm{\alpha t}{u}_t{u}_{\mathrm{xxx}}-\frac{1}{4}\mathrm{\alpha t}{u}_{\mathrm{xx}}{u}_{\mathrm{tx}}+\frac{1}{2}\mathrm{\theta t}{u}_t{u}_x^2+\frac{1}{2}\mathrm{\gamma y}{u}_t{u}_x\\ & +\frac{1}{4}\mathrm{\alpha y}{u}_{\mathrm{xx}}{u}_{\mathrm{xy}}+\frac{1}{4}\mathrm{\alpha y}{u}_{\mathrm{xxx}}{u}_y+\frac{1}{2}\text{β}y{u}_{\mathrm{xx}}{u}_{\mathrm{xz}}\\ & -\frac{1}{2}\mathrm{\theta y}{u}_x^2{u}_z-\frac{1}{4}\mathrm{\alpha z}{u}_{\mathrm{xx}}{u}_{\mathrm{xz}}\\ & +\frac{1}{2}\mathrm{\theta z}{u}_x^2{u}_z+\frac{1}{4}\mathrm{\alpha z}{u}_{\mathrm{xxx}}{u}_z,\end{array}$ $\begin{array}{rl}{C}_8^z& =\frac{1}{4}\text{β}t{u}_t{u}_{\mathrm{xxx}}-\frac{1}{4}\text{β}t{u}_{\mathrm{xx}}{u}_{\mathrm{tx}}+\frac{1}{2}\mathrm{\theta t}{u}_t{u}_x^2+\frac{1}{2}\mathrm{\gamma z}{u}_t{u}_x\\ & -\frac{1}{4}\text{β}y{u}_{\mathrm{xx}}{u}_{\mathrm{xy}}+\frac{1}{4}\text{β}y{u}_{\mathrm{xxx}}{u}_y+\frac{1}{2}\mathrm{\theta y}{u}_x^2{u}_y\\ & +\frac{1}{2}\mathrm{\alpha z}{u}_{\mathrm{xx}}{u}_{\mathrm{xy}}-\frac{1}{2}\mathrm{\theta z}{u}_x^2{u}_y\\ & +\frac{1}{4}\text{β}z{u}_{\mathrm{xx}}{u}_{\mathrm{xz}}+\frac{1}{4}\text{β}z{u}_{\mathrm{xxx}}{u}_z;\\ C_9^t& ={\displaystyle \frac{1}{2}}\mathrm{\gamma u}-\frac{1}{2}\mathrm{x\gamma }{u}_x-\mathrm{t\theta }{u}_z{u}_x-\mathrm{t\theta }{u}_y{u}_x,\\ C_9^x& =\frac{2{\theta }^2}{\gamma }t{u}_z^2{u}_x+\frac{2{\theta }^2}{\gamma }t{u}_y^2{u}_x+\frac{4{\theta }^2}{\gamma }t{u}_z{u}_y{u}_x\\ & -\frac{\text{β}\theta }{\gamma }t{u}_{\mathrm{xz}}^2-\frac{\mathrm{\alpha \theta }}{\gamma }t{u}_{\mathrm{xy}}^2+x{u}_z{u}_x\theta +x{u}_y{u}_x\theta \\ & -\frac{\mathrm{\alpha \theta }}{\gamma }t{u}_{\mathrm{xz}}{u}_{\mathrm{xy}}-\frac{\text{β}\theta }{\gamma }t{u}_{\mathrm{xz}}{u}_{\mathrm{xy}}-\frac{\text{β}\theta }{2\gamma }t{u}_{\mathrm{zz}}{u}_{\mathrm{xx}}\\ & -\frac{\mathrm{\alpha \theta }}{2\gamma }t{u}_{\mathrm{yz}}{u}_{\mathrm{xx}}-\frac{\text{β}\theta }{2\gamma }t{u}_{\mathrm{yz}}{u}_{\mathrm{xx}}-\frac{\mathrm{\alpha \theta }}{2\gamma }t{u}_{\mathrm{yy}}{u}_{\mathrm{xx}}\\ & +\frac{3\text{β}\theta }{2\gamma }t{u}_z{u}_{\mathrm{xxz}}+\frac{3\text{β}\theta }{2\gamma }t{u}_y{u}_{\mathrm{xxz}}+\frac{3\mathrm{\alpha \theta }}{2\gamma }t{u}_z{u}_{\mathrm{xxy}}\\ & +\frac{3\mathrm{\alpha \theta }}{2\gamma }t{u}_y{u}_{\mathrm{xxy}}-t{u}_z{u}_t\theta -t{u}_y{u}_t\theta -\frac{1}{2}\text{β}{u}_{\mathrm{xz}}\\ & -\frac{1}{2}\alpha u_{\mathrm{xy}}+\frac{3}{4}x\text{β}{u}_{\mathrm{xxz}}+\frac{3}{4}\mathrm{x\alpha }{u}_{\mathrm{xxy}}-\frac{1}{2}\mathrm{x\gamma }{u}_t,\\ C_9^y& =\frac{1}{2}\mathrm{x\theta }{u}_x^2+\mathrm{t\theta }{u}_t{u}_x-\frac{1}{4}\alpha u_{\mathrm{xx}}-\frac{\mathrm{\alpha \theta }}{2\gamma }t{u}_{\mathrm{xz}}{u}_{\mathrm{xx}}\\ & +\frac{\text{β}\theta }{\gamma }t{u}_{\mathrm{xz}}{u}_{\mathrm{xx}}+\frac{\mathrm{\alpha \theta }}{2\gamma }t{u}_{\mathrm{xy}}{u}_{\mathrm{xx}}\\ & +\frac{1}{4}\mathrm{x\alpha }{u}_{\mathrm{xxx}}+\frac{\mathrm{\alpha \theta }}{2\gamma }t{u}_z{u}_{\mathrm{xxx}}+\frac{\mathrm{\alpha \theta }}{2\gamma }t{u}_y{u}_{\mathrm{xxx}},\end{array}$ $\begin{array}{rl}{C}_9^z& =\frac{1}{2}\mathrm{x\theta }{u}_x^2+\mathrm{t\theta }{u}_t{u}_x-\frac{1}{4}\text{β}{u}_{\mathrm{xx}}+\frac{\text{β}\theta }{2\gamma }t{u}_{\mathrm{xz}}{u}_{\mathrm{xx}}\\ & +\frac{\mathrm{\alpha \theta }}{\gamma }t{u}_{\mathrm{xy}}{u}_{\mathrm{xx}}-\frac{\text{β}\theta }{2\gamma }t{u}_{\mathrm{xy}}{u}_{\mathrm{xx}}\\ & +\frac{1}{4}x\text{β}{u}_{\mathrm{xxx}}+\frac{\text{β}\theta }{2\gamma }t{u}_z{u}_{\mathrm{xxx}}+\frac{\text{β}\theta }{2\gamma }t{u}_y{u}_{\mathrm{xxx}};\\ C_{10}^t& =-\theta G_2\left(t\right)u_x^2-\frac{1}{2}\mathrm{\gamma y}{G}_2^{\mathrm{\prime }}\left(t\right)u_x-\frac{1}{2}\mathrm{\gamma z}{G}_2^{\mathrm{\prime }}\left(t\right)u_x,\\ C_{10}^x& =\frac{3\mathrm{\alpha \theta }}{2\gamma }{G}_2\left(t\right)u_x{u}_{\mathrm{xxy}}-\frac{\mathrm{\alpha \theta }}{2\gamma }{G}_2\left(t\right)u_{\mathrm{xx}}{u}_{\mathrm{xy}}+\frac{{\theta }^2}{\gamma }{G}_2\left(t\right)u_x^2{u}_y\\ & +\frac{3\text{β}\theta }{2\gamma }{G}_2\left(t\right)u_x{u}_{\mathrm{xxz}}-\frac{\text{β}\theta }{2\gamma }{G}_2\left(t\right)u_{\mathrm{xx}}{u}_{\mathrm{xz}}\\ & +\frac{{\theta }^2}{\gamma }{G}_2\left(t\right)u_x^2{u}_z-\frac{1}{2}\mathrm{\gamma y}{G}_2^{\mathrm{\prime }}\left(t\right)u_t-\frac{1}{2}\mathrm{\gamma z}{G}_2^{\mathrm{\prime }}\left(t\right)u_t\\ & +\frac{3}{4}\mathrm{\alpha z}{G}_2^{\mathrm{\prime }}\left(t\right)u_{\mathrm{xxy}}+\frac{3}{4}\text{β}y{G}_2^{\mathrm{\prime }}\left(t\right)u_{\mathrm{xxz}}+\mathrm{\theta y}{G}_2^{\mathrm{\prime }}\left(t\right)u_x{u}_z\\ & +\mathrm{\theta z}{G}_2^{\mathrm{\prime }}\left(t\right)u_x{u}_y+\frac{3}{4}\mathrm{\alpha y}{G}_2^{\mathrm{\prime }}\left(t\right)u_{\mathrm{xxy}}+\mathrm{\theta y}{G}_2^{\mathrm{\prime }}\left(t\right)u_x{u}_y\\ & +\frac{3}{4}\text{β}z{G}_2^{\mathrm{\prime }}\left(t\right)u_{\mathrm{xxz}}+\mathrm{\theta z}{G}_2^{\mathrm{\prime }}\left(t\right)u_x{u}_z-\frac{1}{4}\alpha G_2^{\mathrm{\prime }}\left(t\right)u_{\mathrm{xx}}\\ & -\frac{1}{4}\text{β}{G}_2^{\mathrm{\prime }}\left(t\right)u_{\mathrm{xx}}+{\displaystyle \frac{1}{2}}\mathrm{\gamma y}{G}_2^{\mathrm{\prime \prime }}\left(t\right)u+{\displaystyle \frac{1}{2}}\mathrm{\gamma z}{G}_2^{\mathrm{\prime \prime }}\left(t\right)u,\\ C_{10}^y& =\frac{\mathrm{\alpha \theta }}{2\gamma }{G}_2\left(t\right)u_{\mathrm{xxx}}{u}_x-\frac{\mathrm{\alpha \theta }}{2\gamma }{G}_2\left(t\right)u_{\mathrm{xx}}^2\\ & +\frac{{\theta }^2}{\gamma }{G}_2\left(t\right)u_x^3+\frac{1}{4}\mathrm{\alpha y}{G}_2^{\mathrm{\prime }}\left(t\right)u_{\mathrm{xxx}}\\ & +\frac{1}{2}\mathrm{\theta y}{G}_2^{\mathrm{\prime }}\left(t\right)u_x^2+\frac{1}{4}\mathrm{\alpha z}{G}_2^{\mathrm{\prime }}\left(t\right)u_{\mathrm{xxx}}+\frac{1}{2}\mathrm{\theta z}{G}_2^{\mathrm{\prime }}\left(t\right)u_x^2,\end{array}$ $\begin{array}{rl}{C}_{10}^z& =\frac{\text{β}\theta }{2\gamma }{G}_2\left(t\right)u_{\mathrm{xxx}}{u}_x-\frac{\text{β}\theta }{2\gamma }{G}_2\left(t\right)u_{\mathrm{xx}}^2\\ & +\frac{{\theta }^2}{\gamma }{G}_2\left(t\right)u_x^3+\frac{1}{4}\text{β}y{G}_2^{\mathrm{\prime }}\left(t\right)u_{\mathrm{xxx}}\\ & +\frac{1}{2}\mathrm{\theta y}{G}_2^{\mathrm{\prime }}\left(t\right)u_x^2+\frac{1}{4}\text{β}z{G}_2^{\mathrm{\prime }}\left(t\right)u_{\mathrm{xxx}}+\frac{1}{2}\mathrm{\theta z}{G}_2^{\mathrm{\prime }}\left(t\right)u_x^2;\\ C_{11}^t& =\frac{{\alpha }^2}{2\text{β}}t{u}_{\mathrm{xx}}{u}_{\mathrm{xy}}-\frac{\mathrm{\alpha \theta }}{2\text{β}}t{u}_x^2{u}_y-\mathrm{\alpha t}{u}_{\mathrm{xx}}{u}_{\mathrm{xy}}\\ & +\mathrm{\theta t}{u}_x^2{u}_y-\frac{\mathrm{\alpha \theta }}{2\text{β}}t{u}_x^2{u}_z+\frac{1}{2}\mathrm{\alpha t}{u}_{\mathrm{xx}}{u}_{\mathrm{xz}}\\ & -\text{β}t{u}_{\mathrm{xx}}{u}_{\mathrm{xz}}+\mathrm{\theta t}{u}_x^2{u}_z-\frac{\mathrm{\alpha \gamma x}{u}_x^2}{2\text{β}}+\frac{1}{2}\mathrm{\gamma x}{u}_x^2\\ & +\frac{\mathrm{\alpha \gamma y}{u}_x{u}_y}{2\text{β}}+\frac{1}{2}\mathrm{\gamma y}{u}_x{u}_y\\ & -\frac{\mathrm{\alpha \gamma z}{u}_x{u}_y}{2\text{β}}+\frac{1}{2}\mathrm{\gamma y}{u}_x{u}_z,\end{array}$ $\begin{array}{rl}{C}_{11}^x& =\frac{{\alpha }^2}{2\text{β}}y{u}_{\mathrm{xy}}^2-\frac{{\alpha }^2}{2\text{β}}z{u}_{\mathrm{xy}}^2-\frac{{\alpha }^2}{2\text{β}}{u}_x{u}_{\mathrm{xy}}+\frac{{\alpha }^2}{4\text{β}}{u}_y{u}_{\mathrm{xx}}\\ & +\frac{{\alpha }^2}{4\text{β}}y{u}_{\mathrm{yy}}{u}_{\mathrm{xx}}-\frac{{\alpha }^2}{4\text{β}}z{u}_{\mathrm{yy}}{u}_{\mathrm{xx}}\\ & -\frac{{\alpha }^2}{4\text{β}}x{u}_{\mathrm{xy}}{u}_{\mathrm{xx}}-\frac{3{\alpha }^2}{4\text{β}}y{u}_y{u}_{\mathrm{xxy}}+\frac{3{\alpha }^2}{4\text{β}}z{u}_y{u}_{\mathrm{xxy}}\\ & +\frac{3{\alpha }^2}{4\text{β}}x{u}_x{u}_{\mathrm{xxy}}+\frac{3{\alpha }^2}{4\text{β}}t{u}_{\mathrm{xxy}}{u}_t\\ & -\frac{{\alpha }^2}{4\text{β}}t{u}_{\mathrm{xx}}{u}_{\mathrm{ty}}-\frac{{\alpha }^2}{2\text{β}}t{u}_{\mathrm{xy}}{u}_{\mathrm{tx}}+\frac{\mathrm{\theta \alpha }}{2\text{β}}x{u}_z{u}_x^2\\ & +\frac{\mathrm{\theta \alpha }}{2\text{β}}x{u}_y{u}_x^2+\frac{1}{2}y{u}_{\mathrm{xy}}^2\alpha -\frac{\mathrm{\gamma \alpha }}{2\text{β}}t{u}_t^2\\ & -\frac{\mathrm{\theta \alpha }}{\text{β}}y{u}_y^2{u}_x+\frac{\mathrm{\theta \alpha }}{\text{β}}z{u}_y^2{u}_x-\frac{\mathrm{\theta \alpha }}{\text{β}}y{u}_z{u}_y{u}_x\\ & +\frac{\mathrm{\theta \alpha }}{\text{β}}z{u}_z{u}_y{u}_x-\frac{1}{2}\alpha u_x{u}_{\mathrm{xz}}\\ & +\frac{1}{2}\alpha u_x{u}_{\mathrm{xy}}+\mathrm{\alpha y}{u}_{\mathrm{xz}}{u}_{\mathrm{xy}}-\frac{1}{2}\mathrm{\alpha z}{u}_{\mathrm{xz}}{u}_{\mathrm{xy}}\\ & +\frac{1}{4}\alpha u_z{u}_{\mathrm{xx}}+\frac{1}{2}\mathrm{\alpha y}{u}_{\mathrm{yz}}{u}_{\mathrm{xx}}\\ & -\frac{1}{4}\mathrm{\alpha z}{u}_{\mathrm{yz}}{u}_{\mathrm{xx}}+\frac{1}{4}\mathrm{\alpha y}{u}_{\mathrm{yy}}{u}_{\mathrm{xx}}-\frac{1}{4}\mathrm{\alpha x}{u}_{\mathrm{xz}}{u}_{\mathrm{xx}}\\ & +\frac{1}{4}\mathrm{\alpha x}{u}_{\mathrm{xy}}{u}_{\mathrm{xx}}-\frac{3}{4}\mathrm{\alpha y}{u}_y{u}_{\mathrm{xxz}}\\ & +\frac{3}{4}\mathrm{\alpha z}{u}_y{u}_{\mathrm{xxz}}+\frac{3}{4}\mathrm{\alpha x}{u}_x{u}_{\mathrm{xxz}}-\frac{3}{4}\mathrm{\alpha y}{u}_z{u}_{\mathrm{xxy}}\\ & -\frac{3}{4}\mathrm{\alpha y}{u}_y{u}_{\mathrm{xxy}}-\frac{3}{4}\mathrm{\alpha x}{u}_x{u}_{\mathrm{xxy}}\\ & +\frac{\mathrm{\gamma \alpha }}{2\text{β}}y{u}_y{u}_t-\frac{\mathrm{\gamma \alpha }}{2\text{β}}z{u}_y{u}_t+\frac{\mathrm{\theta \alpha }}{\text{β}}t{u}_z{u}_x{u}_t\end{array}$ $\begin{array}{rl}& +\frac{\mathrm{\theta \alpha }}{\text{β}}t{u}_y{u}_x{u}_t+\frac{3}{4}\mathrm{\alpha t}{u}_{\mathrm{xxz}}{u}_t\\ & -\frac{3}{2}\mathrm{\alpha t}{u}_{\mathrm{xxy}}{u}_t-\frac{1}{4}\mathrm{\alpha t}{u}_{\mathrm{xx}}{u}_{\mathrm{tz}}+\frac{1}{2}\mathrm{\alpha t}{u}_{\mathrm{xx}}{u}_{\mathrm{ty}}\\ & -\frac{1}{2}\mathrm{\alpha t}{u}_{\mathrm{xz}}{u}_{\mathrm{tx}}+\mathrm{\alpha t}{u}_{\mathrm{xy}}{u}_{\mathrm{tx}}\\ & -\frac{1}{2}\mathrm{\theta x}{u}_z{u}_x^2-\frac{1}{2}\mathrm{\theta x}{u}_y{u}_x^2+\frac{1}{2}\text{β}y{u}_{\mathrm{xz}}^2\\ & +\mathrm{\gamma t}{u}_t^2-\mathrm{\theta y}{u}_z^2{u}_x-\mathrm{\theta y}{u}_y^2{u}_x\\ & -2\mathrm{\theta y}{u}_z{u}_y{u}_x+\frac{1}{2}\text{β}{u}_x{u}_{\mathrm{xz}}+\frac{1}{2}\text{β}y{u}_{\mathrm{xz}}{u}_{\mathrm{xy}}\\ & +\frac{1}{4}\text{β}y{u}_{\mathrm{zz}}{u}_{\mathrm{xx}}+\frac{1}{4}\text{β}y{u}_{\mathrm{yz}}{u}_{\mathrm{xx}}\\ & +\frac{1}{4}\text{β}x{u}_{\mathrm{xz}}{u}_{\mathrm{xx}}-\frac{3}{4}\text{β}y{u}_z{u}_{\mathrm{xxz}}-\frac{3}{4}\text{β}y{u}_y{u}_{\mathrm{xxz}}\\ & -\frac{3}{4}\text{β}x{u}_x{u}_{\mathrm{xxz}}+\frac{1}{2}\mathrm{\gamma y}{u}_z{u}_t\\ & +\frac{1}{2}\mathrm{\gamma y}{u}_y{u}_t-2\mathrm{\theta t}{u}_z{u}_x{u}_t-2\mathrm{\theta t}{u}_y{u}_x{u}_t\\ & -\frac{3}{2}\text{β}t{u}_{\mathrm{xxz}}{u}_t+\frac{1}{2}\text{β}t{u}_{\mathrm{xx}}{u}_{\mathrm{tz}}+\text{β}t{u}_{\mathrm{xz}}{u}_{\mathrm{tx}},\\ C_{11}^y& =-\frac{1}{2}\mathrm{\theta x}{u}_x^3+\frac{\mathrm{\alpha \theta }}{2\text{β}}x{u}_x^3+\frac{\mathrm{\alpha \theta }}{2\text{β}}y{u}_z{u}_x^2-\frac{\mathrm{\alpha \theta }}{2\text{β}}z{u}_z{u}_x^2\\ & -\mathrm{\theta t}{u}_t{u}_x^2+\frac{\mathrm{\alpha \theta }}{2\text{β}}t{u}_t{u}_x^2\\ & +\frac{1}{4}\alpha u_{\mathrm{xx}}{u}_x-\frac{{\alpha }^2}{4\text{β}}{u}_{\mathrm{xx}}{u}_x-\frac{1}{4}\mathrm{\alpha x}{u}_{\mathrm{xxx}}{u}_x\\ & +\frac{{\alpha }^2}{4\text{β}}x{u}_{\mathrm{xxx}}{u}_x-\frac{1}{2}\mathrm{\gamma y}{u}_t{u}_x\end{array}$ $\begin{array}{rl}& -\frac{\mathrm{\alpha \gamma }}{2\text{β}}y{u}_t{u}_x+\frac{\mathrm{\alpha \gamma }}{2\text{β}}z{u}_t{u}_x+\frac{1}{4}\mathrm{\alpha x}{u}_{\mathrm{xx}}^2-\frac{{\alpha }^2}{4\text{β}}x{u}_{\mathrm{xx}}^2\\ & -\frac{1}{4}\mathrm{\alpha y}{u}_{\mathrm{xz}}{u}_{\mathrm{xx}}+\frac{1}{2}\mathrm{\alpha z}{u}_{\mathrm{xz}}{u}_{\mathrm{xx}}-\frac{1}{2}\text{β}y{u}_{\mathrm{xz}}{u}_{\mathrm{xx}}\\ & -\frac{1}{4}\mathrm{\alpha y}{u}_{\mathrm{xy}}{u}_{\mathrm{xx}}-\frac{{\alpha }^2}{4\text{β}}y{u}_{\mathrm{xy}}{u}_{\mathrm{xx}}+\frac{{\alpha }^2}{4\text{β}}z{u}_{\mathrm{xy}}{u}_{\mathrm{xx}}\\ & -\frac{1}{4}\mathrm{\alpha y}{u}_z{u}_{\mathrm{xxx}}-\frac{1}{4}\mathrm{\alpha y}{u}_y{u}_{\mathrm{xxx}}-\frac{{\alpha }^2}{4\text{β}}y{u}_y{u}_{\mathrm{xxx}}\\ & +\frac{{\alpha }^2}{4\text{β}}z{u}_y{u}_{\mathrm{xxx}}-\frac{1}{2}\mathrm{\alpha t}{u}_{\mathrm{xxx}}{u}_t\\ & +\frac{{\alpha }^2}{4\text{β}}t{u}_{\mathrm{xxx}}{u}_t+\frac{1}{2}\mathrm{\alpha t}{u}_{\mathrm{xx}}{u}_{\mathrm{tx}}-\frac{{\alpha }^2}{4\text{β}}t{u}_{\mathrm{xx}}{u}_{\mathrm{tx}},\\ C_{11}^z& =\frac{\mathrm{\alpha \theta }}{2\text{β}}t{u}_t{u}_x^2+\frac{1}{4}\mathrm{\alpha t}{u}_t{u}_{\mathrm{xxx}}-\frac{1}{4}\mathrm{\alpha t}{u}_{\mathrm{xx}}{u}_{\mathrm{tx}}\\ & -\frac{1}{2}\text{β}t{u}_t{u}_{\mathrm{xxx}}+\frac{1}{2}\text{β}t{u}_{\mathrm{xx}}{u}_{\mathrm{tx}}\\ & -\mathrm{\theta t}{u}_t{u}_x^2-\frac{1}{2}\mathrm{\gamma y}{u}_t{u}_x+\frac{\mathrm{\alpha \theta }}{2\text{β}}x{u}_x^3-\frac{1}{4}\alpha u_{\mathrm{xx}}{u}_x\\ & +\frac{1}{4}\mathrm{\alpha x}{u}_{\mathrm{xxx}}{u}_x-\frac{1}{4}\mathrm{\alpha x}{u}_{\mathrm{xx}}^2\\ & +\frac{1}{4}\text{β}{u}_{\mathrm{xx}}{u}_x-\frac{1}{4}\text{β}x{u}_{\mathrm{xxx}}{u}_x+\frac{1}{4}\text{β}x{u}_{\mathrm{xx}}^2\\ & -\frac{1}{2}\mathrm{\theta x}{u}_x^3-\frac{\mathrm{\alpha \theta }}{2\text{β}}y{u}_x^2{u}_y-\frac{1}{4}\mathrm{\alpha y}{u}_{\mathrm{xx}}{u}_{\mathrm{xy}}\\ & -\frac{1}{4}\mathrm{\alpha y}{u}_{\mathrm{xxx}}{u}_y+\frac{1}{4}\text{β}y{u}_{\mathrm{xx}}{u}_{\mathrm{xy}}-\frac{1}{4}\text{β}y{u}_{\mathrm{xxx}}{u}_y\\ & +\frac{\mathrm{\alpha \theta }}{2\text{β}}z{u}_x^2{u}_y-\frac{1}{4}\mathrm{\alpha z}{u}_{\mathrm{xx}}{u}_{\mathrm{xy}}\\ & +\frac{1}{4}\mathrm{\alpha z}{u}_{\mathrm{xxx}}{u}_y-\frac{1}{4}\text{β}y{u}_{\mathrm{xx}}{u}_{\mathrm{xz}}-\frac{1}{4}\text{β}y{u}_{\mathrm{xxx}}{u}_z.\end{array}$

4.1.1. Application of the conserved quantities related to Noether's symmetries in physics and engineering

Conservation laws are revealed to be fundamental laws of nature. Generally speaking, the cumulative quantity of property controlled by that law remains unaltered in the course of physical processes. Regarding classical physics, conservation laws consist of the conservation of energy, linear momentum, mass (or matter), electric charge and angular momentum. Having said that, the integration of a given non-trivial conservation law over the spatial domain given as $\mathrm{\Omega }\subseteq \mathbb{R}$ purveys a conserved integral of the form [21] (87) $\mathbf{T}\left[u\right]={{\int }_{\mathrm{\Omega }}C\mathrm{d}x|}_{\mathcal{E}},$(87) where we have $\mathcal{E}$ stands for the space of solutions of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ), which is presented by the equation as well as its differential consequences embedded in the jet space.

We now present the corresponding conserved quantities, which are known to have a variety of meanings and uses in the physical sciences, in relation to some of the conservation laws that have been achieved. Relative to vector $(C_1^t,C_1^x,C_1^y,C_1^z)$ furnishes (88) $\begin{array}{rl}{\mathbf{T}}_1\left[u\right]& ={\int }_{\mathrm{\Omega }}(\frac{1}{2}\alpha u_{\mathrm{xx}}{u}_{\mathrm{xy}}-\frac{1}{2}\theta u_x^2{u}_y\\ & +\frac{1}{2}\text{β}{u}_{\mathrm{xx}}{u}_{\mathrm{xz}}-\frac{1}{2}\theta u_x^2{u}_z)\mathrm{d}x,\end{array}$(88) which represents the conserved quantity of energy. Meanwhile, with regard to $(C_i^t,C_i^x,C_i^y,C_i^z,i=2,3,4)$, we have the conserved relations (89) $\begin{array}{rl}{\mathbf{T}}_2\left[u\right]& =-{\int }_{\mathrm{\Omega }}\frac{1}{2}\gamma u_x^2\mathrm{d}x,{\mathbf{T}}_3\left[u\right]=-{\int }_{\mathrm{\Omega }}\frac{1}{2}\gamma u_x{u}_y\mathrm{d}x,\\ & \mathrm{and}{\mathbf{T}}_4\left[u\right]=-{\int }_{\mathrm{\Omega }}\frac{1}{2}\gamma u_x{u}_z\mathrm{d}x\end{array}$(89) preserving conservation of momenta. Related to $(C_i^t,C_i^x,C_i^y,C_i^z,i=5,6)$, one has (90) $\begin{array}{rl}{\mathbf{T}}_5\left[u\right]& ={\int }_{\mathrm{\Omega }}\frac{1}{2}\gamma G_1\left(t\right)u_x\mathrm{d}x,\mathrm{and}\\ {\mathbf{T}}_6\left[u\right]& ={\int }_{\mathrm{\Omega }}\frac{1}{2}\gamma G_1(t,z-y)u_x\mathrm{d}x\end{array}$(90) representing the generalized momenta of model (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ). Next, we examine conserved vectors $(C_7^t,C_7^x,C_7^y,C_7^z)$ and $(C_8^t,C_8^x,C_8^y,C_8^z)$, corresponding to these, we have (91) $\begin{array}{rl}{\mathbf{T}}_7\left[u\right]& ={\int }_{\mathrm{\Omega }}(\phantom{\frac{1}{2}}\mathrm{\alpha t}{u}_{\mathrm{xx}}{u}_{\mathrm{xy}}-\mathrm{\theta t}{u}_x^2{u}_y+\text{β}t{u}_{\mathrm{xx}}{u}_{\mathrm{xz}}-\mathrm{\theta t}{u}_x^2{u}_z\\ & -\frac{1}{2}\mathrm{\gamma u}{u}_x-\frac{1}{2}\mathrm{\gamma x}{u}_x^2)\mathrm{d}x,\\ {\mathbf{T}}_8\left[u\right]& ={\int }_{\mathrm{\Omega }}(\frac{1}{2}\mathrm{\alpha t}{u}_{\mathrm{xx}}{u}_{\mathrm{xy}}-\frac{1}{2}\mathrm{\theta t}{u}_x^2{u}_y+\frac{1}{2}\text{β}t{u}_{\mathrm{xx}}{u}_{\mathrm{xz}}\\ & -\frac{1}{2}\mathrm{\theta t}{u}_x^2{u}_z-\frac{1}{2}\mathrm{\gamma y}{u}_x{u}_y-\frac{1}{2}\mathrm{\gamma z}{u}_x{u}_z)\mathrm{d}x,\end{array}$(91) which represent conservations of energy. Similarly, the other conservation laws represent mathematically conserved quantities if one adheres to the previously described pattern. In mechanical engineering, the conservation of energy is applied to design efficient machines and systems. For instance, in the design of engines and turbines, understanding how energy is conserved and transferred allows engineers to optimize performance. Besides, for scientific applications in physics, the conservation of energy is fundamental. It is applied in fields such as thermodynamics, which governs heat transfer and works in various processes. In particle physics, the conservation of momentum is essential for understanding the behaviour of subatomic particles in collisions. In aerospace engineering, the conservation of momentum is crucial in understanding the motion of rockets and spacecraft. Engineers use this principle to design control systems for accurate manoeuvring [80].

5. Concluding remarks

In this research paper, we examined a generalized nonlinear multi-dimensional breaking soliton equation (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) where various classical solutions for the model are secured via the theory of the Lie group. For the very first time, the generalized equation was comprehensively studied via Lie group approach which occasions a detailed reduction of the said equation using its Lie point symmetries. The implementation of the technique assisted us in gaining an ordinary differential equation which is integrable to secure a solution to the underlying equation. Furthermore, the simplest equation technique with regard to Bernoulli and Riccati equations was utilized. As a result, we constructed soliton solutions consisting of elliptic integral function, topological kink wave, algebraic, logarithmic, dark-bright and singular solutions. The featuring of the elliptic function solution gave the most general solution of the equation which disintegrates under some certain limits to produce hyperbolic and various other elementary trigonometric function solutions. In addition, we discussed the solutions as well as the graphical demonstration of the secured results which were presented to view their streaming patterns. These graphical representations showcase appreciable wave profiles of the solutions. They include dark–bright, bright, dark, singular and kink waves. Lastly, we constructed the conserved quantities of (5(5) $\begin{array}{rl}& \alpha u_{\mathrm{xxxy}}+\text{β}{u}_{\mathrm{xxxz}}-\gamma u_{\mathrm{xt}}+2\theta u_x{u}_{\mathrm{xy}}+2\theta u_x{u}_{\mathrm{xz}}\\ & +\theta u_y{u}_{\mathrm{xx}}+\theta u_z{u}_{\mathrm{xx}}=0,\end{array}$(5) ) by utilizing the Noether theorem which occasions 10 local conserved vectors of the equation under study. Furthermore, we outlined various significance and applications of the obtained conservation laws as well as exact solutions which included exploration of the shape and we also visualize it to see how it related to the real-world problem. These, as a result, made the work in this paper novel and original. The results secured in this paper may be of particular interest to scientists working in various fields of science and engineering, especially in physics, marine science and ocean engineering, for their analysis because of their applicability. The future scope of this work includes the opportunity of engaging the obtained conservation laws to find more solutions to the model using double reduction technique.

Authors' contributions

All authors wrote the manuscript.

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Additional information

Funding

The authors did not receive funding from any organization for this work.