Lie group analysis, solitons, self-adjointness and conservation laws of the nonlinear elastic structural element equation

Abstract

This study is based on the Lie group method for the nonlinear elastic structural element equation (ESE Equation). We obtain a three-dimensional Lie algebra. By utilizing this Lie algebra a four-dimensional optimal system is constructed. The governing ESE Equation is converted to nonlinear ordinary differential equations (ODEs) via symmetry reduction. We use a modified auxiliary equation (MAE) procedure to deal with nonlinear ODEs. These ODEs reveal the dynamics of the periodic and soliton solutions. We obtain soliton solutions through rational, trigonometric, and hyperbolic functions. Wolfram Mathematica simulations vividly illustrate the wave characteristics of the derived solutions, affirming their properties as singular periodic solutions, a singular solution, an optical dark soliton solution, and a singular soliton solution. We also obtain the local conservation laws by a new conservation theorem introduced by Ibragimov.

Keywords:

• Lie group method
• optimal system
• nonlinear elastic structural element equation
• Lie algebra
• modified auxiliary equation procedure

1. Introduction

In the present research of nonlinear complex physical processes, exact solutions for the nonlinear evolution equations (NEEs) are extremely important. Numerous novel techniques have been proposed in recent years, including the tanh-method [1], the Jacobi elliptic function method [2], the homotopy perturbation approach [3], the homogeneous balancing method [4], the sine-cosine method [5], the simplest equation technique [6], the modified auxiliary equation method [7], the improved F-expansion technique [8], the exp-function approach [9], the ($G^{\prime }/G$)-expansion technique [10], the Hirota bilinear technique [11], the new extended direct algebraic method [12], and others [13]. More importantly, Lie initially suggested the Lie group analysis, which was afterward studied by Ovsiannikov [14], Ibragimov [15], Bluman [16], Olver [17,18], and Stephani [19]. The nonlinear Vakhnenko-Parkes equation, pseudo-parabolic equations, nonlinear wave equations in elasticity, reduced micro morphic model, Riabouchinsky Proudman Johnson equation, generalized Pochhammer-Charee equation, the Thomas equation [20], Lonngren equation, Schrödinger equation and many other nonlinear equations were successfully and fruitfully solved using this method.

The widely recognized Korteweg-de Vries (KdV) model, describing the dimensionless fluid depth $w=w(x,t)$ of long surface waves on shallow water, is expressed as follows (1) $\begin{array}{r}{w}_t+\alpha w{w}_x+\gamma w_{xxx}=0,\end{array}$(1) where α is parameter of nonlinearity and γ is a dispersion parameter. The adjusted version of this model, commonly referred to as the modified KdV model or equation, is formulated as follows (2) $\begin{array}{r}{w}_t+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(2) where β is parameter of nonlinearity and γ is a dispersion parameter. There exists a substantial body of literature, and a few notable references are cited here [21–31], addressing the KdV and modified KdV equations. Our focus centres on concurrently investigating the influence of both models. To this end, we introduce a nonlinear model that scrutinizes the combined effects of both (1(1) $\begin{array}{r}{w}_t+\alpha w{w}_x+\gamma w_{xxx}=0,\end{array}$(1) ) and (2(2) $\begin{array}{r}{w}_t+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(2) ), as expressed by (3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) where $\alpha ,\beta$ are parameters of nonlinearity, γ is a dispersion parameter and $w(x,t)$ is the displacement function. Referred to as the nonlinear elastic structural element (ESE) equation, it emerges in elasticity and various other domains. When $\beta =0$, Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) corresponds to the KdV model (1(1) $\begin{array}{r}{w}_t+\alpha w{w}_x+\gamma w_{xxx}=0,\end{array}$(1) ), and in the case of $\alpha =0$, it represents the modified KdV model (2(2) $\begin{array}{r}{w}_t+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(2) ). Our discussion centres on this model from the perspective of closed-form solutions. The novelty in our approach lies in applying the group analysis method for obtaining closed-form solutions to the nonlinear ESE Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ). We further present soliton solutions grounded in hyperbolic, trigonometric, and rational functions. Notably, we showcase singular periodic solutions, a singular solution, an optical dark soliton solution, and a singular soliton solution for the ESE Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ).

The contents of the paper are organized as follows: The symmetry finding approach for a nonlinear ESE Equation that supplies the base elements for the optimal system is described in Section 2. For the symmetry reductions in Section 3, we use the optimal system. We define the MAE procedure algorithm in Section 4 to deal with soliton solutions with the aid of governing ODEs. In Section 5, we discuss the self-adjointness of the ESE Equation and obtain the conservation laws using Ibragimov's theorem. The generated soliton wave patterns' dynamics are demonstrated with numerical evaluation and discussion utilizing symbolic computation in Section 6. The final thoughts on the study are presented in Section 7, along with a few novel concepts for further research.

2. Lie classification and optimal system

Following is a brief summary of the Lie symmetry [14]. Consider a one-parameter Lie group of transformation (4) $\begin{array}{r}\begin{array}{rl}& \tilde{x}\to x+\varsigma {\psi }_1(x,t,w)+O({\varsigma }^2),\\ & \tilde{t}\to t+\varsigma {\psi }_2(x,t,w)+O({\varsigma }^2),\\ & \tilde{w}\to w+\varsigma \eta (x,t,w)+O({\varsigma }^2),\end{array}\end{array}$(4) with ς acting as the group parameter. The vector field that corresponds to the transformations mentioned above is defined as (5) $\begin{array}{r}\mathrm{\aleph }={\psi }_1(x,t,w)\frac{\mathrm{\partial }}{\mathrm{\partial }x}+{\psi }_2(x,t,w)\frac{\mathrm{\partial }}{\mathrm{\partial }t}+\eta (x,t,w)\frac{\mathrm{\partial }}{\mathrm{\partial }w}.\end{array}$(5) Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) requires the third order prolonged generator ${\mathrm{\aleph }}^{[3]}$ given by (6) $\begin{array}{r}{\mathrm{\aleph }}^{[3]}=\mathrm{\aleph }+\sum _{s=1}^3{\eta }_{i_1\cdots i_s}\frac{\mathrm{\partial }}{\mathrm{\partial }{w}_{i_1\cdots i_s}},\end{array}$(6) where ${\eta }_{i_1\cdots i_s}=D_{i_s}{\eta }_{i_1\dots i_{s-1}}-(D_{i_s}{\psi }_l)w_{i_1,\dots ,i_{k-1}l}$, and $D_i$ is a total derivative operator.

Then by invariance condition [15] of Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) we get (7) $\begin{array}{r}{\mathrm{\aleph }}^{[3]}(w_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}){|}_{(3)}=0.\end{array}$(7) Separating different polynomials in w and its derivatives, condition (7(7) $\begin{array}{r}{\mathrm{\aleph }}^{[3]}(w_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}){|}_{(3)}=0.\end{array}$(7) ) gives rise to an overdetermined system of PDEs given by (8) $\begin{array}{r}\begin{array}{rl}& {{\psi }_2}_{tt}=0,{{\psi }_2}_w=0,{{\psi }_2}_x=0,\\ & {{\psi }_1}_t+\frac{{\alpha }^2}{6\beta }{{\psi }_2}_t=0,{{\psi }_1}_w=0,\\ & {{\psi }_1}_x-\frac{{{\psi }_2}_t}{3}=0,{\eta }_w+\frac{(\alpha +2\beta w)}{6\beta }{{\psi }_2}_t=0.\end{array}\end{array}$(8) The solution of system (8(8) $\begin{array}{r}\begin{array}{rl}& {{\psi }_2}_{tt}=0,{{\psi }_2}_w=0,{{\psi }_2}_x=0,\\ & {{\psi }_1}_t+\frac{{\alpha }^2}{6\beta }{{\psi }_2}_t=0,{{\psi }_1}_w=0,\\ & {{\psi }_1}_x-\frac{{{\psi }_2}_t}{3}=0,{\eta }_w+\frac{(\alpha +2\beta w)}{6\beta }{{\psi }_2}_t=0.\end{array}\end{array}$(8) ) provides the infinitesimals given by (9) $\begin{array}{rl}{\psi }_1& =\frac{c_1}{3}x-\frac{c_1{\alpha }^2}{6\beta }t+c_3,\\ {\psi }_2& =c_{1}t+c_2,\\ \eta & =-\frac{c_1(\alpha +2\beta w)}{6\beta },\end{array}$(9) where $c_1,c_2,c_3$ are arbitrary constants. From Equation (9(9) $\begin{array}{rl}{\psi }_1& =\frac{c_1}{3}x-\frac{c_1{\alpha }^2}{6\beta }t+c_3,\\ {\psi }_2& =c_{1}t+c_2,\\ \eta & =-\frac{c_1(\alpha +2\beta w)}{6\beta },\end{array}$(9) ), we get the following three symmetry generators (10) $\begin{array}{r}\begin{array}{rl}{\mathrm{\aleph }}_1& =\frac{\mathrm{\partial }}{\mathrm{\partial }t},\\ {\mathrm{\aleph }}_2& =\frac{\mathrm{\partial }}{\mathrm{\partial }x},\\ {\mathrm{\aleph }}_3& =({\alpha }^{2}t-2\beta x)\frac{\mathrm{\partial }}{\mathrm{\partial }x}-6\beta t\frac{\mathrm{\partial }}{\mathrm{\partial }t}+(2\beta w+\alpha )\frac{\mathrm{\partial }}{\mathrm{\partial }w}.\end{array}\end{array}$(10) The Lie algebra formed by these symmetry generators under Lie bracket $[\cdot ,\cdot ]$ has the following commutator Table 1. Also, the adjoint representation [17] is shown via Table 2.

Table 1. Commutator table.

$[{\mathrm{\aleph }}_i,{\mathrm{\aleph }}_j]$	${\mathrm{\aleph }}_1$	${\mathrm{\aleph }}_2$	${\mathrm{\aleph }}_3$
${\mathrm{\aleph }}_1$	0	0	${\alpha }^2{\mathrm{\aleph }}_2-6\beta {\mathrm{\aleph }}_1$
${\mathrm{\aleph }}_2$	0	0	$-2\beta {\mathrm{\aleph }}_2$
${\mathrm{\aleph }}_3$	$-{\alpha }^2{\mathrm{\aleph }}_2+6\beta {\mathrm{\aleph }}_1$	$2\beta {\mathrm{\aleph }}_2$	0

Table 2. Adjoint table.

$Ad(e^{ϵ})$	${\mathrm{\aleph }}_1$	${\mathrm{\aleph }}_2$	${\mathrm{\aleph }}_3$
${\mathrm{\aleph }}_1$	${\mathrm{\aleph }}_1$	${\mathrm{\aleph }}_2$	${\mathrm{\aleph }}_3+6\beta ϵ{\mathrm{\aleph }}_1-{\alpha }^2ϵ{\mathrm{\aleph }}_2$
${\mathrm{\aleph }}_2$	${\mathrm{\aleph }}_1$	${\mathrm{\aleph }}_2$	${\mathrm{\aleph }}_3+2\beta {\mathrm{\aleph }}_2$
${\mathrm{\aleph }}_3$	$\begin{array}{l}{e}^{-6\beta ϵ}{\mathrm{\aleph }}_1-\frac{{\alpha }^2}{4\beta }{e}^{-2\beta ϵ}\\ \times (e^{-4\beta ϵ}-1){\mathrm{\aleph }}_2\end{array}$	$e^{-2\beta ϵ}{\mathrm{\aleph }}_2$	${\mathrm{\aleph }}_3$

Theorem 2.1

[18]

Let ${\mathrm{L}}^3$ be the Lie algebra given in Equation (10(10) $\begin{array}{r}\begin{array}{rl}{\mathrm{\aleph }}_1& =\frac{\mathrm{\partial }}{\mathrm{\partial }t},\\ {\mathrm{\aleph }}_2& =\frac{\mathrm{\partial }}{\mathrm{\partial }x},\\ {\mathrm{\aleph }}_3& =({\alpha }^{2}t-2\beta x)\frac{\mathrm{\partial }}{\mathrm{\partial }x}-6\beta t\frac{\mathrm{\partial }}{\mathrm{\partial }t}+(2\beta w+\alpha )\frac{\mathrm{\partial }}{\mathrm{\partial }w}.\end{array}\end{array}$(10) ). In this case, the optimal system of subalgebras with only one-dimension can be discovered using the generators listed below. (11) $\begin{array}{r}\begin{array}{rl}{\mathcal{O}}_1& =⟨{\mathrm{\aleph }}_3⟩,\\ {\mathcal{O}}_2& =⟨{\mathrm{\aleph }}_2⟩,\\ {\mathcal{O}}_3& =⟨{\mathrm{\aleph }}_1⟩,\\ {\mathcal{O}}_4& =⟨{\mathrm{\aleph }}_1\pm {\mathrm{\aleph }}_2⟩.\end{array}\end{array}$(11)

Proof.

Let us take an arbitrary element $\mathrm{\aleph }\in {\mathrm{L}}^3.$ We have, (12) $\begin{array}{r}\mathrm{\aleph }=x_1{\mathrm{\aleph }}_1+x_2{\mathrm{\aleph }}_2+x_3{\mathrm{\aleph }}_3\end{array}$(12) where $x_i,i=1,2,3$ are any real parameters.

Case 1: $x_3\ne 0$. Then we have, (13) $\begin{array}{rl}\mathrm{\aleph }& =x_1{\mathrm{\aleph }}_1+x_2{\mathrm{\aleph }}_2+x_3{\mathrm{\aleph }}_3\end{array}$(13) (14) $\begin{array}{rl}{\mathrm{\aleph }}^{\mathrm{\prime }}& =Ad(e^{ϵ}{\mathrm{\aleph }}_1)\mathrm{\aleph }=x_3{\mathrm{\aleph }}_3\end{array}$(14) Hence, we get (15) $\begin{array}{r}{\mathcal{O}}_1={\mathrm{\aleph }}_3,\end{array}$(15) Case 2: $x_3=0,x_1=0$. Then we have, (16) $\begin{array}{r}\mathrm{\aleph }=x_2{\mathrm{\aleph }}_2\end{array}$(16) Hence, we get (17) $\begin{array}{r}{\mathcal{O}}_2={\mathrm{\aleph }}_2,\end{array}$(17) Case 3: $x_3=0,x_1\ne 0,x_2=0$. Then we have, (18) $\begin{array}{r}\mathrm{\aleph }=x_1{\mathrm{\aleph }}_1\end{array}$(18) Hence, we get (19) $\begin{array}{r}{\mathcal{O}}_3={\mathrm{\aleph }}_1,\end{array}$(19) Case 4: $x_3=0,x_1\ne 0,x_2\ne 0$. Take $x_1=1.$ Then we have, (20) $\begin{array}{rl}\mathrm{\aleph }& ={\mathrm{\aleph }}_1+x_2{\mathrm{\aleph }}_2\end{array}$(20) (21) $\begin{array}{rl}{\mathrm{\aleph }}^{\mathrm{\prime }}& =Ad(e^{ϵ}{\mathrm{\aleph }}_3)\mathrm{\aleph }\\ & ={\mathrm{\aleph }}_1+(x_2{e}^{4\beta ϵ}-\frac{{\alpha }^2}{4\beta }{e}^{4\beta ϵ}(e^{-4\beta ϵ}-1)){\mathrm{\aleph }}_2\end{array}$(21) Hence, we get (22) $\begin{array}{r}{\mathcal{O}}_4={\mathrm{\aleph }}_1\pm {\mathrm{\aleph }}_2.\end{array}$(22)

3. Symmetry reductions via subalgebras

3.1. Reduction by ${\mathcal{O}}_2=⟨{\mathrm{\aleph }}_2⟩$

Through the application of the characteristic approach, we adhere to the characteristic equation relevant to the vector field ${\mathrm{\aleph }}_2$ written as $\begin{array}{r}\frac{\mathrm{d}x}{1}=\frac{\mathrm{d}t}{0}=\frac{\mathrm{d}w}{0}.\end{array}$After solving the above-mentioned equation, we acquire the similarity variables $w=h(r),r=t$. By employing this transformation, it becomes feasible to express the simplified form of Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) as follows $h^{\prime }=0,$ with solution $h(r)=c_1.$ Consequently, the invariant solution of Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) in the main variables can be expressed as (23) $\begin{array}{r}w(x,t)=c_1.\end{array}$(23)

3.2. Reduction by ${\mathcal{O}}_3=⟨{\mathrm{\aleph }}_1⟩$

Through the application of the characteristic approach, we adhere to the characteristic equation relevant to the vector field ${\mathrm{\aleph }}_1$ written as $\begin{array}{r}\frac{\mathrm{d}x}{0}=\frac{\mathrm{d}t}{1}=\frac{\mathrm{d}w}{0}.\end{array}$After solving the above-mentioned equation, we acquire the similarity variables $w=h(r),r=x$ and the reduced ODE becomes (24) $\begin{array}{r}\gamma h^{‴}+(\alpha +\beta h)h{h}^{\prime }=0.\end{array}$(24) Acknowledging the complexity of solving this nonlinear ordinary differential equation, our focus will shift to exploring soliton solutions for the ESE Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) by examining the corresponding ODE (24(24) $\begin{array}{r}\gamma h^{‴}+(\alpha +\beta h)h{h}^{\prime }=0.\end{array}$(24) ).

3.3. Reduction by ${\mathcal{O}}_1=⟨{\mathrm{\aleph }}_3⟩$

Through the application of the characteristic approach, we adhere to the characteristic equation relevant to the vector field ${\mathrm{\aleph }}_3$ written as $\begin{array}{r}\frac{\mathrm{d}x}{{\alpha }^{2}t-2\beta x}=\frac{\mathrm{d}t}{-6\beta t}=\frac{\mathrm{d}w}{2\beta w+\alpha }.\end{array}$After solving the above-mentioned equation, we acquire the similarity variables $w=\frac{2\beta h(r)-\alpha t^{\frac{1}{3}}}{2\beta t^{\frac{1}{3}}},r=\frac{{\alpha }^{2}t+4\beta x}{4\beta t^{\frac{1}{3}}},$ with reduced ODE given by (25) $\begin{array}{r}3\gamma h^{‴}+(3\beta h^2-r)h^{\prime }-h=0.\end{array}$(25) We only follow the trivial solution of the above ODE. Therefore, we get the solution (26) $\begin{array}{r}w(x,t)=-\frac{\alpha }{2\beta }.\end{array}$(26)

3.4. Reduction by ${\mathcal{O}}_4=⟨{\mathrm{\aleph }}_1+{\mathrm{\aleph }}_2⟩$

Through the application of the characteristic approach, we adhere to the characteristic equation relevant to the vector field ${\mathrm{\aleph }}_1+{\mathrm{\aleph }}_2$ written as $\begin{array}{r}\frac{\mathrm{d}x}{1}=\frac{\mathrm{d}t}{1}=\frac{\mathrm{d}w}{0}.\end{array}$After solving the above-mentioned equation, we acquire the similarity variables $w=k(r),r=-x+t$. By employing this transformation, it becomes feasible to express the simplified form of Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) as follows (27) $\begin{array}{r}\gamma h^{‴}+\beta h^2{h}^{\prime }+\alpha h{h}^{\prime }-h^{\prime }=0.\end{array}$(27) Integrating the above equation and taking the constant of integration to be zero (28) $\begin{array}{r}6\gamma h^{″}+2\beta h^3+3\alpha h^2-6h=0.\end{array}$(28) We also deal with the solution of this ODE in the next section.

3.5. Reduction by ${\mathcal{O}}_4=⟨{\mathrm{\aleph }}_1-{\mathrm{\aleph }}_2⟩$

Through the application of the characteristic approach, we adhere to the characteristic equation relevant to the vector field ${\mathrm{\aleph }}_1-{\mathrm{\aleph }}_2$ written as $\begin{array}{r}\frac{\mathrm{d}x}{-1}=\frac{\mathrm{d}t}{1}=\frac{\mathrm{d}w}{0}.\end{array}$The similarity variables becomes $w=k(r),r=x+t$ and the reduced ODE is (29) $\begin{array}{r}\gamma h^{‴}+\beta h^2{h}^{\prime }+\alpha h{h}^{\prime }+h^{\prime }=0.\end{array}$(29) Integrating the above equation and taking the constant of integration to be zero (30) $\begin{array}{r}6\gamma h^{″}+2\beta h^3+3\alpha h^2+6h=0.\end{array}$(30) In the next section, we introduce the algorithm for the MAE procedure to deal with the soliton solutions of the governing ODEs (24(24) $\begin{array}{r}\gamma h^{‴}+(\alpha +\beta h)h{h}^{\prime }=0.\end{array}$(24) ), (28(28) $\begin{array}{r}6\gamma h^{″}+2\beta h^3+3\alpha h^2-6h=0.\end{array}$(28) ) and (30(30) $\begin{array}{r}6\gamma h^{″}+2\beta h^3+3\alpha h^2+6h=0.\end{array}$(30) ).

4. Description of the MAE procedure

The soliton solutions are very helpful in investigating the dynamical behaviour and nature of the physical phenomenon. Here we use to describe the algorithm for the MAE procedure [32] to use for the soliton solutions for the reduced ODEs (24(24) $\begin{array}{r}\gamma h^{‴}+(\alpha +\beta h)h{h}^{\prime }=0.\end{array}$(24) ), (28(28) $\begin{array}{r}6\gamma h^{″}+2\beta h^3+3\alpha h^2-6h=0.\end{array}$(28) ) and (30(30) $\begin{array}{r}6\gamma h^{″}+2\beta h^3+3\alpha h^2+6h=0.\end{array}$(30) ). Here are the main steps of the algorithm.

Step 1: We take into account an NPDE in its basic form (31) $\begin{array}{r}{E}_1(w,w_t,w_x,w_{xx},\dots ,)=0.\end{array}$(31) Step 2: We use the transform $w(x,t)=h(r),r=x-ct$ to Equation (31(31) $\begin{array}{r}{E}_1(w,w_t,w_x,w_{xx},\dots ,)=0.\end{array}$(31) ), to get the following ODE (32) $\begin{array}{r}{E}_2(h,h^{\prime },h^{″},\dots )=0.\end{array}$(32) For the MAE procedure, we follow the following ansatz solution (33) $\begin{array}{r}h(r)=a_0+\sum _{\varsigma =1}^{\varphi }[a_{\varsigma }{\left(T^h\right)}^{\varsigma }+b_{\varsigma }{\left(T^h\right)}^{-\varsigma }],\end{array}$(33) where T, $a_{\varsigma }$, $b_{\varsigma }$, and $a_0$ are arbitrary constants, and $h(r)$ fulfils the auxiliary equation (34) $\begin{array}{r}{h}^{\mathrm{\prime }}(r)=\frac{{\mathrm{\Psi }}_1+{\mathrm{\Psi }}_2{T}^{-h}+{\mathrm{\Psi }}_3{T}^h}{\ln (T)},\end{array}$(34) where ${\mathrm{\Psi }}_1$, ${\mathrm{\Psi }}_2$, and ${\mathrm{\Psi }}_3$ are arbitrary constants and $T>0,T\ne 1$. The solution of the auxiliary Equation (34(34) $\begin{array}{r}{h}^{\mathrm{\prime }}(r)=\frac{{\mathrm{\Psi }}_1+{\mathrm{\Psi }}_2{T}^{-h}+{\mathrm{\Psi }}_3{T}^h}{\ln (T)},\end{array}$(34) ) is given below taking into account different cases.

Case i: If $4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3-{\mathrm{\Psi }}_3^2<0,{\mathrm{\Psi }}_3\ne 0$, then $\begin{array}{rl}{T}^h& =\frac{-{\mathrm{\Psi }}_2+\sqrt{4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3-{\mathrm{\Psi }}_3^2}\tan \left(\frac{\sqrt{4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3-{\mathrm{\Psi }}_3^2}\zeta }{2}\right)}{2{\mathrm{\Psi }}_3}\text{ or}\\ T^h& =-\frac{{\mathrm{\Psi }}_2+\sqrt{4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3-{\mathrm{\Psi }}_3^2}\cot \left(\frac{\sqrt{4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3-{\mathrm{\Psi }}_3^2}\zeta }{2}\right)}{2{\mathrm{\Psi }}_3}\end{array}$Case ii: If ${\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3>0,{\mathrm{\Psi }}_3\ne 0$, then $\begin{array}{rl}{T}^h& =\frac{-{\mathrm{\Psi }}_2+\sqrt{{\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3}\tanh \left(\frac{\sqrt{{\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3}\zeta }{2}\right)}{2{\mathrm{\Psi }}_3}\text{ or}\\ T^h& =-\frac{{\mathrm{\Psi }}_2+\sqrt{{\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3}\coth \left(\frac{\sqrt{{\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3}\zeta }{2}\right)}{2{\mathrm{\Psi }}_3}\end{array}$Case iii: If ${\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3=0$, and ${\mathrm{\Psi }}_3\ne 0$, then $\begin{array}{r}{T}^h=-\frac{2+\beta \zeta }{2{\mathrm{\Psi }}_3\zeta }.\end{array}$Step 3: In this algorithm step, the homogeneous balancing principle is used to indicate the balancing number ϕ.

Step 4: For the solution of Equation (31(31) $\begin{array}{r}{E}_1(w,w_t,w_x,w_{xx},\dots ,)=0.\end{array}$(31) ), we back substitute the values in Equation (32(32) $\begin{array}{r}{E}_2(h,h^{\prime },h^{″},\dots )=0.\end{array}$(32) ) and then replace the transform defined in step 2.

Now we utilize the above-defined algorithm of the MAE procedure to the nonlinear elastic structural element Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ).

4.1. Soliton solutions of the ESE Equation (3)

By using the homogeneous balancing principle on Equation (24(24) $\begin{array}{r}\gamma h^{‴}+(\alpha +\beta h)h{h}^{\prime }=0.\end{array}$(24) ), we get the balancing number $\varphi =1$. Then by Equation (33(33) $\begin{array}{r}h(r)=a_0+\sum _{\varsigma =1}^{\varphi }[a_{\varsigma }{\left(T^h\right)}^{\varsigma }+b_{\varsigma }{\left(T^h\right)}^{-\varsigma }],\end{array}$(33) ) we get (35) $\begin{array}{r}h(r)=a_0+a_1{T}^{h(r)}+b_1{T}^{-h(r)}.\end{array}$(35) We insert the values of the required expression in Equation (24(24) $\begin{array}{r}\gamma h^{‴}+(\alpha +\beta h)h{h}^{\prime }=0.\end{array}$(24) ) and then apply Equation (34(34) $\begin{array}{r}{h}^{\mathrm{\prime }}(r)=\frac{{\mathrm{\Psi }}_1+{\mathrm{\Psi }}_2{T}^{-h}+{\mathrm{\Psi }}_3{T}^h}{\ln (T)},\end{array}$(34) ). Then by comparing different powers of $T^h$, we obtain an algebraic system. To handle this system we use computing tools like Mathematica or Maple to get the following solution sets.

Group (1): $\begin{array}{rl}\alpha & =\frac{6\gamma {\mathrm{\Psi }}_1(2a_0{\mathrm{\Psi }}_1)-b_1{\mathrm{\Psi }}_2}{b_1^2},a_0=a_0,\\ a_1& =\frac{6a_0{\mathrm{\Psi }}_1^2-6a_0{b}_1{\mathrm{\Psi }}_1{\mathrm{\Psi }}_2+b_1^2{\mathrm{\Psi }}_2^2}{4b_1{\mathrm{\Psi }}_1^2},b_1=b_1,\\ \beta & =-\frac{6\gamma {\mathrm{\Psi }}_1^2}{b_1^2}.\end{array}$Inserting back the parameters values of Group 1 to Equation (24(24) $\begin{array}{r}\gamma h^{‴}+(\alpha +\beta h)h{h}^{\prime }=0.\end{array}$(24) ), we get the following categories of the soliton solutions;

Strategy (1): When $\mathrm{\Delta }={\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3<0$, and ${\mathrm{\Psi }}_3\ne 0$, we have the solution of Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) in the form of trigonometric functions is (36) $\begin{array}{rl}{w}_1(x,t)& =a_0+\frac{6a_0{\mathrm{\Psi }}_1^2-6a_0{b}_1{\mathrm{\Psi }}_1{\mathrm{\Psi }}_2+b_1^2{\mathrm{\Psi }}_2^2}{4b_1{\mathrm{\Psi }}_1^2}\\ & \times \{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\tan \left(\frac{\sqrt{-\mathrm{\Delta }}x}{2}\right)\}\\ & +b_1{\{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\tan \left(\frac{\sqrt{-\mathrm{\Delta }}x}{2}\right)\}}^{-1},\end{array}$(36) or (37) $\begin{array}{rl}{w}_2(x,t)& =a_0+\frac{6a_0{\mathrm{\Psi }}_1^2-6a_0{b}_1{\mathrm{\Psi }}_1{\mathrm{\Psi }}_2+b_1^2{\mathrm{\Psi }}_2^2}{4b_1{\mathrm{\Psi }}_1^2}\\ & \times \{-\frac{{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\cot \left(\frac{\sqrt{-\mathrm{\Delta }}x}{2}\right)\}\\ & +b_1{\{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\cot \left(\frac{\sqrt{-\mathrm{\Delta }}x}{2}\right)\}}^{-1}.\end{array}$(37) Strategy (2): When $\mathrm{\Delta }={\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3>0$, and ${\mathrm{\Psi }}_3\ne 0$, the solution of Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) in the form of hyperbolic functions is (38) $\begin{array}{rl}{w}_3(x,t)& =a_0+\frac{6a_0{\mathrm{\Psi }}_1^2-6a_0{b}_1{\mathrm{\Psi }}_1{\mathrm{\Psi }}_2+b_1^2{\mathrm{\Psi }}_2^2}{4b_1{\mathrm{\Psi }}_1^2}\\ & \times \{\frac{-{\mathrm{\Psi }}_2+\sqrt{\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\tanh \left(\frac{\sqrt{\mathrm{\Delta }}x}{2}\right)\}\\ & +b_1{\{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\tan \left(\frac{\sqrt{\mathrm{\Delta }}x}{2}\right)\}}^{-1},\end{array}$(38) or (39) $\begin{array}{rl}{w}_4(x,t)& =a_0+\frac{6a_0{\mathrm{\Psi }}_1^2-6a_0{b}_1{\mathrm{\Psi }}_1{\mathrm{\Psi }}_2+b_1^2{\mathrm{\Psi }}_2^2}{4b_1{\mathrm{\Psi }}_1^2}\\ & \times \{-\frac{{\mathrm{\Psi }}_2+\sqrt{\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\tan \left(\frac{\sqrt{\mathrm{\Delta }}x}{2}\right)\}\\ & +b_1{\{\frac{-{\mathrm{\Psi }}_2+\sqrt{\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\coth \left(\frac{\sqrt{\mathrm{\Delta }}x}{2}\right)\}}^{-1}.\end{array}$(39) Strategy (3): When $\mathrm{\Delta }={\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3=0$, and ${\mathrm{\Psi }}_3\ne 0$, we have the following rational form of the solution; (40) $\begin{array}{rl}{w}_5(x,t)& =a_0+\frac{6a_0{\mathrm{\Psi }}_1^2-6a_0{b}_1{\mathrm{\Psi }}_1{\mathrm{\Psi }}_2+b_1^2{\mathrm{\Psi }}_2^2}{4b_1{\mathrm{\Psi }}_1^2}\\ & \times \{-\frac{2+\beta x}{2{\mathrm{\Psi }}_{3}x}\}+b_1{\{-\frac{2+\beta x}{2{\mathrm{\Psi }}_{3}x}\}}^{-1}.\end{array}$(40) Group (2): $\begin{array}{rl}\alpha & =\frac{6\gamma {\mathrm{\Psi }}_1(2a_0{\mathrm{\Psi }}_1)-b_1{\mathrm{\Psi }}_2}{b_1^2},a_0=a_0,a_1=0,\\ b_1& =b_1,\beta =-\frac{6\gamma {\mathrm{\Psi }}_1^2}{b_1^2}.\end{array}$Inserting back the values of the parameters of Group 2 to Equation (24(24) $\begin{array}{r}\gamma h^{‴}+(\alpha +\beta h)h{h}^{\prime }=0.\end{array}$(24) ), we get the following categories of the soliton solutions;

Strategy (4): When $\mathrm{\Delta }={\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3<0$, and ${\mathrm{\Psi }}_3\ne 0$, we have the solution of Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) in the form of trigonometric functions is (41) $\begin{array}{r}{w}_6(x,t)=a_0+b_1{\{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\tan \left(\frac{\sqrt{-\mathrm{\Delta }}x}{2}\right)\}}^{-1},\end{array}$(41) or (42) $\begin{array}{r}{w}_7(x,t)=a_0+b_1{\{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\cot \left(\frac{\sqrt{-\mathrm{\Delta }}x}{2}\right)\}}^{-1}.\end{array}$(42) Strategy (5): When $\mathrm{\Delta }={\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3>0$, and ${\mathrm{\Psi }}_3\ne 0$, the solution of Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) in the form of hyperbolic functions is (43) $\begin{array}{r}{w}_8(x,t)=a_0+b_1{\{\frac{-{\mathrm{\Psi }}_2+\sqrt{\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\tanh \left(\frac{\sqrt{\mathrm{\Delta }}x}{2}\right)\}}^{-1},\end{array}$(43) or (44) $\begin{array}{r}{w}_9(x,t)=a_0+b_1{\{\frac{-{\mathrm{\Psi }}_2+\sqrt{\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\coth \left(\frac{\sqrt{\mathrm{\Delta }}x}{2}\right)\}}^{-1}.\end{array}$(44) Strategy (6): When $\mathrm{\Delta }={\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3=0$, and ${\mathrm{\Psi }}_3\ne 0$, we have the following rational form of the solution; (45) $\begin{array}{r}{w}_{10}(x,t)=a_0+b_1{\{-\frac{2+\beta x}{2{\mathrm{\Psi }}_{3}x}\}}^{-1}.\end{array}$(45) Group (3): $\begin{array}{rl}\alpha & =\frac{6\gamma {\mathrm{\Psi }}_3(2a_0{\mathrm{\Psi }}_3)-a_1{\mathrm{\Psi }}_2}{a_1^2},a_0=a_0,a_1=a_1,\\ b_1& =0,\beta =-\frac{6\gamma {\mathrm{\Psi }}_3^2}{a_1^2}.\end{array}$Inserting back the values of parameters of Group 3 to Equation (24(24) $\begin{array}{r}\gamma h^{‴}+(\alpha +\beta h)h{h}^{\prime }=0.\end{array}$(24) ), we get the following categories of the soliton solutions;

Strategy (7): When $\mathrm{\Delta }={\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3<0$, and ${\mathrm{\Psi }}_3\ne 0$, we have the solution of Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) in the form of trigonometric functions is (46) $\begin{array}{r}{w}_{11}(x,t)=a_0+a_1\{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\tan \left(\frac{\sqrt{-\mathrm{\Delta }}x}{2}\right)\},\end{array}$(46) or (47) $\begin{array}{r}{w}_{12}(x,t)=a_0+a_1\{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\cot \left(\frac{\sqrt{-\mathrm{\Delta }}x}{2}\right)\}.\end{array}$(47) Strategy (8): When $\mathrm{\Delta }={\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3>0$, and ${\mathrm{\Psi }}_3\ne 0$, the solution of Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) in the form of hyperbolic functions is (48) $\begin{array}{r}{w}_{13}(x,t)=a_0+a_1\{\frac{-{\mathrm{\Psi }}_2+\sqrt{\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\tanh \left(\frac{\sqrt{\mathrm{\Delta }}x}{2}\right)\},\end{array}$(48) or (49) $\begin{array}{r}{w}_{14}(x,t)=a_0+a_1\{\frac{-{\mathrm{\Psi }}_2+\sqrt{\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\coth \left(\frac{\sqrt{\mathrm{\Delta }}x}{2}\right)\}.\end{array}$(49) Strategy (9): When $\mathrm{\Delta }={\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3=0$, and ${\mathrm{\Psi }}_3\ne 0$, we have the following rational form of the solution; (50) $\begin{array}{r}{w}_{15}(x,t)=a_0+a_1\{-\frac{2+\beta x}{2{\mathrm{\Psi }}_{3}x}\}.\end{array}$(50) Now we apply the algorithm of MAE procedure to the reduced ODE (28(28) $\begin{array}{r}6\gamma h^{″}+2\beta h^3+3\alpha h^2-6h=0.\end{array}$(28) ) to obtain the soliton solutions for the nonlinear ESE Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ). By using the homogeneous balancing principle on Equation (28(28) $\begin{array}{r}6\gamma h^{″}+2\beta h^3+3\alpha h^2-6h=0.\end{array}$(28) ), we get the balancing number $\varphi =1$. Then by Equation (33(33) $\begin{array}{r}h(r)=a_0+\sum _{\varsigma =1}^{\varphi }[a_{\varsigma }{\left(T^h\right)}^{\varsigma }+b_{\varsigma }{\left(T^h\right)}^{-\varsigma }],\end{array}$(33) ) we get (51) $\begin{array}{r}h(r)=a_0+a_1{T}^{h(r)}+b_1{T}^{-h(r)}.\end{array}$(51) We insert the values of the required expression in Equation (28(28) $\begin{array}{r}6\gamma h^{″}+2\beta h^3+3\alpha h^2-6h=0.\end{array}$(28) ) and then apply Equation (34(34) $\begin{array}{r}{h}^{\mathrm{\prime }}(r)=\frac{{\mathrm{\Psi }}_1+{\mathrm{\Psi }}_2{T}^{-h}+{\mathrm{\Psi }}_3{T}^h}{\ln (T)},\end{array}$(34) ). Then by comparing different powers of the $T^h$, we obtain an algebraic system. To handle this system we use computing tools like Mathematica or Maple to have the following solution sets.

Group (1): $\begin{array}{rl}\alpha & =\frac{6\gamma {\mathrm{\Psi }}_1(2a_0{\mathrm{\Psi }}_1)-b_1{\mathrm{\Psi }}_2}{b_1^2}\\ a_1& =\frac{6\gamma a_0^2{\mathrm{\Psi }}_1^2-6\gamma a_0{b}_1{\mathrm{\Psi }}_1{\mathrm{\Psi }}_2+\gamma b_1^2{\mathrm{\Psi }}_2^2-b^2-1}{4\gamma b_1{\mathrm{\Psi }}_1^2}\\ \beta & =-\frac{6\gamma {\mathrm{\Psi }}_1^2}{b_1^2}.\end{array}$Inserting back the parameters values of Group 1 to Equation (28(28) $\begin{array}{r}6\gamma h^{″}+2\beta h^3+3\alpha h^2-6h=0.\end{array}$(28) ), we get the following categories of the soliton solutions;

Strategy (10): When $\mathrm{\Delta }={\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3<0$, and ${\mathrm{\Psi }}_3\ne 0$, we have the solution of Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) in the form of trigonometric functions is (52) $\begin{array}{rl}{w}_{16}(x,t)& =a_0+\frac{\begin{array}{l}6\gamma a_0^2{\mathrm{\Psi }}_1^2-6\gamma a_0{b}_1{\mathrm{\Psi }}_1{\mathrm{\Psi }}_2\\ +\gamma b_1^2{\mathrm{\Psi }}_2^2-b^2-1\end{array}}{4\gamma b_1{\mathrm{\Psi }}_1^2}\\ & \times \{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\tan \left(\frac{\sqrt{-\mathrm{\Delta }}(t-x)}{2}\right)\}\\ & +b_1{\{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\tan \left(\frac{\sqrt{-\mathrm{\Delta }}x}{2}\right)\}}^{-1},\end{array}$(52) or (53) $\begin{array}{rl}{w}_{17}(x,t)& =a_0+\frac{\begin{array}{l}6\gamma a_0^2{\mathrm{\Psi }}_1^2-6\gamma a_0{b}_1{\mathrm{\Psi }}_1{\mathrm{\Psi }}_2\\ +\gamma b_1^2{\mathrm{\Psi }}_2^2-b^2-1\end{array}}{4\gamma b_1{\mathrm{\Psi }}_1^2}\\ & \times \{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\cot \left(\frac{\sqrt{-\mathrm{\Delta }}x}{2}\right)\}\\ & +b_1{\{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\tan \left(\frac{\sqrt{-\mathrm{\Delta }}(t-x)}{2}\right)\}}^{-1}.\end{array}$(53) Strategy (11): When $\mathrm{\Delta }={\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3>0$, and ${\mathrm{\Psi }}_3\ne 0$, the solution of Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) in the form of hyperbolic functions is (54) $\begin{array}{rl}{w}_{18}(x,t)& =a_0+\frac{\begin{array}{l}6\gamma a_0^2{\mathrm{\Psi }}_1^2-6\gamma a_0{b}_1{\mathrm{\Psi }}_1{\mathrm{\Psi }}_2\\ +\gamma b_1^2{\mathrm{\Psi }}_2^2-b^2-1\end{array}}{4\gamma b_1{\mathrm{\Psi }}_1^2}\\ & \times \{\frac{-{\mathrm{\Psi }}_2+\sqrt{\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\tanh \left(\frac{\sqrt{\mathrm{\Delta }}x}{2}\right)\}\\ & +b_1{\{\frac{-{\mathrm{\Psi }}_2+\sqrt{\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\tanh \left(\frac{\sqrt{\mathrm{\Delta }}(t-x)}{2}\right)\}}^{-1},\end{array}$(54) or (55) $\begin{array}{rl}{w}_{19}(x,t)& =a_0+\frac{\begin{array}{l}6\gamma a_0^2{\mathrm{\Psi }}_1^2-6\gamma a_0{b}_1{\mathrm{\Psi }}_1{\mathrm{\Psi }}_2\\ +\gamma b_1^2{\mathrm{\Psi }}_2^2-b^2-1\end{array}}{4\gamma b_1{\mathrm{\Psi }}_1^2}\\ & \times \{\frac{-{\mathrm{\Psi }}_2+\sqrt{\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\coth \left(\frac{\sqrt{\mathrm{\Delta }}x}{2}\right)\}\\ & +b_1{\{\frac{-{\mathrm{\Psi }}_2+\sqrt{\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\coth \left(\frac{\sqrt{\mathrm{\Delta }}(t-x)}{2}\right)\}}^{-1}.\end{array}$(55) Strategy (12): When $\mathrm{\Delta }={\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3=0$, and ${\mathrm{\Psi }}_3\ne 0$, we have the following rational form of the solution; (56) $\begin{array}{rl}{w}_{20}(x,t)& =a_0+\frac{\begin{array}{l}6\gamma a_0^2{\mathrm{\Psi }}_1^2-6\gamma a_0{b}_1{\mathrm{\Psi }}_1{\mathrm{\Psi }}_2\\ +\gamma b_1^2{\mathrm{\Psi }}_2^2-b^2-1\end{array}}{4\gamma b_1{\mathrm{\Psi }}_1^2}\\ & \times \{-\frac{2+\beta x}{2{\mathrm{\Psi }}_{3}x}\}+b_1{\{-\frac{2+\beta (t-x)}{2{\mathrm{\Psi }}_3(t-x)}\}}^{-1}.\end{array}$(56) The second set Group 2 of parameters is given by $\begin{array}{rl}\alpha & =\frac{6\gamma {\mathrm{\Psi }}_1(2a_0{\mathrm{\Psi }}_1)-b_1{\mathrm{\Psi }}_2}{b_1^2},a_0=a_0,a_1=0,\\ b_1& =b_1,\beta =-\frac{6\gamma {\mathrm{\Psi }}_1^2}{b_1^2}.\end{array}$Inserting back the values of the parameters of Group 2 to Equation (28(28) $\begin{array}{r}6\gamma h^{″}+2\beta h^3+3\alpha h^2-6h=0.\end{array}$(28) ), we get the following categories of the soliton solutions;

Strategy (13): When $\mathrm{\Delta }={\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3<0$, and ${\mathrm{\Psi }}_3\ne 0$, we have the solution of Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) in the form of trigonometric functions is (57) $\begin{array}{rl}{w}_{21}(x,t)& =a_0+b_1\{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\\ & \times {\tan \left(\frac{\sqrt{-\mathrm{\Delta }}(t-x)}{2}\right)\}}^{-1},\end{array}$(57) or (58) $\begin{array}{rl}{w}_{22}(x,t)& =a_0+b_1\{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\\ & \times {\cot \left(\frac{\sqrt{-\mathrm{\Delta }}(t-x)}{2}\right)\}}^{-1}.\end{array}$(58) Strategy (14): When $\mathrm{\Delta }={\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3>0$, and ${\mathrm{\Psi }}_3\ne 0$, the solution of Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) in the form of hyperbolic functions is (59) $\begin{array}{rl}{w}_{23}(x,t)& =a_0+b_1\{\frac{-{\mathrm{\Psi }}_2+\sqrt{\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\\ & \times {\tanh \left(\frac{\sqrt{\mathrm{\Delta }}(t-x)}{2}\right)\}}^{-1},\end{array}$(59) or (60) $\begin{array}{rl}{w}_{24}(x,t)& =a_0+b_1\{\frac{-{\mathrm{\Psi }}_2+\sqrt{\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\\ & \times {\coth \left(\frac{\sqrt{\mathrm{\Delta }}(t-x)}{2}\right)\}}^{-1}.\end{array}$(60) Strategy (15): When $\mathrm{\Delta }={\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3=0$, and ${\mathrm{\Psi }}_3\ne 0$, we have the following rational form of the solution; (61) $\begin{array}{rl}{w}_{25}(x,t)& =a_0+b_1{\{-\frac{2+\beta (t-x)}{2{\mathrm{\Psi }}_3(t-x)}\}}^{-1}.\end{array}$(61) The third set Group 3 of parameters is $\begin{array}{rl}\alpha & =\frac{6\gamma {\mathrm{\Psi }}_3(2a_0{\mathrm{\Psi }}_3)-a_1{\mathrm{\Psi }}_2}{a_1^2},a_0=a_0,a_1=a_1,\\ b_1& =0,\beta =-\frac{6\gamma {\mathrm{\Psi }}_3^2}{a_1^2}.\end{array}$Inserting back the values of parameters of Group 3 to Equation (28(28) $\begin{array}{r}6\gamma h^{″}+2\beta h^3+3\alpha h^2-6h=0.\end{array}$(28) ), we get the following categories of the soliton solutions;

Strategy (16): When $\mathrm{\Delta }={\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3<0$, and ${\mathrm{\Psi }}_3\ne 0$, we have the solution of Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) in the form of trigonometric functions is (62) $\begin{array}{rl}{w}_{26}(x,t)& =a_0+a_1\{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\\ & \times \tan \left(\frac{\sqrt{-\mathrm{\Delta }}(t-x)}{2}\right)\},\end{array}$(62) or (63) $\begin{array}{rl}{w}_{27}(x,t)& =a_0+a_1\{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\\ & \times \cot \left(\frac{\sqrt{-\mathrm{\Delta }}(t-x)}{2}\right)\}.\end{array}$(63) Strategy (17): When $\mathrm{\Delta }={\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3>0$, and ${\mathrm{\Psi }}_3\ne 0$, the solution of Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) in the form of hyperbolic functions is (64) $\begin{array}{rl}{w}_{28}(x,t)& =a_0+a_1\{\frac{-{\mathrm{\Psi }}_2+\sqrt{\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\\ & \times \tanh \left(\frac{\sqrt{\mathrm{\Delta }}(t-x)}{2}\right)\},\end{array}$(64) or (65) $\begin{array}{rl}{w}_{29}(x,t)& =a_0+a_1\{\frac{-{\mathrm{\Psi }}_2+\sqrt{\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\\ & \times \coth \left(\frac{\sqrt{\mathrm{\Delta }}(t-x)}{2}\right)\}.\end{array}$(65) Strategy (18): When $\mathrm{\Delta }={\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3=0$, and ${\mathrm{\Psi }}_3\ne 0$, we have the following rational form of the solution; (66) $\begin{array}{r}{w}_{30}(x,t)=a_0+a_1\{-\frac{2+\beta (t-x)}{2{\mathrm{\Psi }}_3(t-x)}\}.\end{array}$(66) For the more categories of the soliton solutions, we will now proceed with the MAE procedure to the reduced ODE (30(30) $\begin{array}{r}6\gamma h^{″}+2\beta h^3+3\alpha h^2+6h=0.\end{array}$(30) ). We follow the same steps for this ODE (30(30) $\begin{array}{r}6\gamma h^{″}+2\beta h^3+3\alpha h^2+6h=0.\end{array}$(30) ) and have a set of values of parameters. For the solutions via the MAE procedure algorithm, we have the set of parameter values given by Group 1 $\begin{array}{rl}& C=-\frac{1}{{\mathrm{\Psi }}_1\beta }\sqrt{\begin{array}{l}-\frac{1}{1875\gamma \beta }(-16{\alpha }^3\sqrt{-6{\alpha }^2+45\beta }+92{\alpha }^4\\ +60\alpha \beta \sqrt{-6{\alpha }^2+45\beta }-690{\alpha }^2\beta \\ +1350{\beta }^2)\end{array}}\\ & a_0=-\frac{1}{75{\mathrm{\Psi }}_1\beta }\sqrt{-\frac{\begin{array}{l}48{\alpha }^3\sqrt{-6{\alpha }^2+45\beta }+276{\alpha }^4\\ -180\alpha \beta \sqrt{-6{\alpha }^2+45\beta }\\ -2070{\alpha }^2\beta +4050{\beta }^2\end{array}}{\gamma \beta }},\\ & a_1=a_1,b_1=b_1,\beta =-\frac{{\gamma }^2{\mathrm{\Psi }}_2^4+2\gamma {\mathrm{\Psi }}_2^2+1}{b_1{a}_1\gamma {\mathrm{\Psi }}_2^2}.\end{array}$Inserting back the parameters values of Group 1 to Equation (30(30) $\begin{array}{r}6\gamma h^{″}+2\beta h^3+3\alpha h^2+6h=0.\end{array}$(30) ), we get the following three categories of the soliton solutions;

Strategy (19): When $\mathrm{\Delta }={\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3<0$, and ${\mathrm{\Psi }}_3\ne 0$, we have the solution of Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) in the form of trigonometric functions is (67) $\begin{array}{rl}& w_{31}(x,t)\\ & =-\frac{1}{75{\mathrm{\Psi }}_1\beta }\sqrt{-\frac{\begin{array}{l}48{\alpha }^3\sqrt{-6{\alpha }^2+45\beta }+276{\alpha }^4\\ -180\alpha \beta \sqrt{-6{\alpha }^2+45\beta }\\ -2070{\alpha }^2\beta +4050{\beta }^2\end{array}}{\gamma \beta }}\\ & +a_1\{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\tan \left(\frac{\sqrt{-\mathrm{\Delta }}x}{2}\right)\}\\ & +b_1{\{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\tan \left(\frac{\sqrt{-\mathrm{\Delta }}(x+t)}{2}\right)\}}^{-1},\end{array}$(67) or (68) $\begin{array}{rl}& w_{32}(x,t)\\ & =-\frac{1}{75{\mathrm{\Psi }}_1\beta }\sqrt{-\frac{\begin{array}{l}48{\alpha }^3\sqrt{-6{\alpha }^2+45\beta }+276{\alpha }^4\\ -180\alpha \beta \sqrt{-6{\alpha }^2+45\beta }\\ -2070{\alpha }^2\beta +4050{\beta }^2\end{array}}{\gamma \beta }}\\ & +a_1\{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\cot \left(\frac{\sqrt{-\mathrm{\Delta }}x}{2}\right)\}\\ & +b_1{\{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\cot \left(\frac{\sqrt{-\mathrm{\Delta }}(x+t)}{2}\right)\}}^{-1}.\end{array}$(68) Strategy (20): When $\mathrm{\Delta }={\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3>0$, and ${\mathrm{\Psi }}_3\ne 0$, the solution of Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) in the form of hyperbolic functions is (69) $\begin{array}{rl}& w_{33}(x,t)\\ & =-\frac{1}{75{\mathrm{\Psi }}_1\beta }\sqrt{-\frac{\begin{array}{l}48{\alpha }^3\sqrt{-6{\alpha }^2+45\beta }+276{\alpha }^4\\ -180\alpha \beta \sqrt{-6{\alpha }^2+45\beta }\\ -2070{\alpha }^2\beta +4050{\beta }^2\end{array}}{\gamma \beta }}\\ & +a_1\{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\tanh \left(\frac{\sqrt{-\mathrm{\Delta }}x}{2}\right)\}\\ & +b_1{\{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\tanh \left(\frac{\sqrt{-\mathrm{\Delta }}(x+t)}{2}\right)\}}^{-1},\end{array}$(69) or (70) $\begin{array}{rl}& w_{34}(x,t)\\ & =-\frac{1}{75{\mathrm{\Psi }}_1\beta }\sqrt{-\frac{\begin{array}{l}48{\alpha }^3\sqrt{-6{\alpha }^2+45\beta }+276{\alpha }^4\\ -180\alpha \beta \sqrt{-6{\alpha }^2+45\beta }\\ -2070{\alpha }^2\beta +4050{\beta }^2\end{array}}{\gamma \beta }}\\ & +a_1\{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\coth \left(\frac{\sqrt{-\mathrm{\Delta }}x}{2}\right)\}\\ & +b_1{\{\frac{-{\mathrm{\Psi }}_2+\sqrt{-\mathrm{\Delta }}}{2{\mathrm{\Psi }}_3}\coth \left(\frac{\sqrt{-\mathrm{\Delta }}(x+t)}{2}\right)\}}^{-1}.\end{array}$(70) Strategy (21): When $\mathrm{\Delta }={\mathrm{\Psi }}_2^2-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3=0$, and ${\mathrm{\Psi }}_3\ne 0$, we have the following rational form of the solution; (71) $\begin{array}{rl}& w_{35}(x,t)\\ & =-\frac{1}{75{\mathrm{\Psi }}_1\beta }\sqrt{-\frac{\begin{array}{l}48{\alpha }^3\sqrt{-6{\alpha }^2+45\beta }+276{\alpha }^4\\ -180\alpha \beta \sqrt{-6{\alpha }^2+45\beta }\\ -2070{\alpha }^2\beta +4050{\beta }^2\end{array}}{\gamma \beta }}\\ & +a_1\{-\frac{2+\beta x}{2{\mathrm{\Psi }}_3(x+t)}\}+b_1{\{-\frac{2+\beta x}{2{\mathrm{\Psi }}_3(x+t)}\}}^{-1}.\end{array}$(71)

5. Self-adjointness and conservation laws

In this section, we compute the conservation laws for the nonlinear ESE Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) by using the new conservation theorem of Ibragimov [33]. For this purpose, we first consider a nth order PDE (72) $\begin{array}{r}E(x,w,w_{(1)},w_{(2)},\dots ,w_{(n)})=0,\end{array}$(72) where $w=w(x)$ and $x=x(x_1,x_2,\dots ,x_m)$. Also, $w_{(1)}$ and $w_{(n)}$ denote the first and nth derivatives of w.

Theorem 5.1

[33,34]

For a given system (72(72) $\begin{array}{r}E(x,w,w_{(1)},w_{(2)},\dots ,w_{(n)})=0,\end{array}$(72) ) along with the adjoint equation (73) $\begin{array}{r}{E}^{\star }(x,w,v,w_{(1)},v_{(1)},w_{(2)},v_{(2)}\cdots ,w_{(n)},v_{(n)})=0,\end{array}$(73) there exists a formal Lagrangian given by $\begin{array}{r}\mathcal{L}=vE(x,w,w_{(1)},w_{(2)},\dots ,w_{(n)})=0.\end{array}$Moreover, the adjoint Equation (73(73) $\begin{array}{r}{E}^{\star }(x,w,v,w_{(1)},v_{(1)},w_{(2)},v_{(2)}\cdots ,w_{(n)},v_{(n)})=0,\end{array}$(73) ) is defined by (74) $\begin{array}{r}{E}^{\star }\equiv \frac{\delta }{\delta w}(vE),\end{array}$(74) and the operator $\frac{\delta }{\delta w}$ is known as Euler-Lagrange operator given by (75) $\begin{array}{r}\frac{\delta }{\delta w}=\frac{\mathrm{\partial }}{\mathrm{\partial }w}+\sum _{i=1}^{\mathrm{\infty }}(-1{)}^s{D}_{i^1}\cdots D_{i^s}\frac{\mathrm{\partial }}{\mathrm{\partial }{w}_{i^1\cdots i^s}},\end{array}$(75) where $D_i$ is a total derivative operator given by (76) $\begin{array}{r}{D}_i=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}+w_i\frac{\mathrm{\partial }}{\mathrm{\partial }w}+w_{ij}\frac{\mathrm{\partial }}{\mathrm{\partial }{w}_j}+\dots .\end{array}$(76)

Definition 5.1

Equation (72(72) $\begin{array}{r}E(x,w,w_{(1)},w_{(2)},\dots ,w_{(n)})=0,\end{array}$(72) ) is said to be a strictly self-adjoint if the equation obtained from its adjoint equation with the help of the transformation v = w, is given by (77) $\begin{array}{r}{E}^{\star }{|}_{v=w}=\mu (x,w,\dots )E,\end{array}$(77) for some $x\in \mathcal{D}$ ($\mathcal{D}$ is a particular domain).

Definition 5.2

Equation (72(72) $\begin{array}{r}E(x,w,w_{(1)},w_{(2)},\dots ,w_{(n)})=0,\end{array}$(72) ) is said to be a quasi-self-adjoint if the equation obtained from its adjoint equation with the help of the transformation $v=v(w)\ne 0$, is given by (78) $\begin{array}{r}{E}^{\star }{|}_{v=v(w)}=\mu (x,w,\dots )E,\end{array}$(78) where $x\in \mathcal{D}$.

Definition 5.3

Equation (72(72) $\begin{array}{r}E(x,w,w_{(1)},w_{(2)},\dots ,w_{(n)})=0,\end{array}$(72) ) is said to be a weak self-adjoint if the equation acquired from its adjoint equation under the transformation $v=v(x,w)\ne 0$ for a particular function v such that $v_w\ne 0$ and $v_{x^i}\ne 0$ for some $x^i$, such that (79) $\begin{array}{r}{E}^{\star }{|}_{v=v(x,w)}=\mu (x,w,\dots )E,\end{array}$(79) for some $x\in \mathcal{D}$.

Definition 5.4

Equation (72(72) $\begin{array}{r}E(x,w,w_{(1)},w_{(2)},\dots ,w_{(n)})=0,\end{array}$(72) ) is said to be nonlinearly self-adjoint under the substitution $v=v(x,w)$ if the equation obtained from its adjoint equation, with some function such that $v(x,w)\ne 0$, (72(72) $\begin{array}{r}E(x,w,w_{(1)},w_{(2)},\dots ,w_{(n)})=0,\end{array}$(72) ) satisfy the condition, (80) $\begin{array}{r}{E}^{\star }{|}_{v=v(x,w)}=\mu (x,w,\dots )E,\end{array}$(80) for some $x\in \mathcal{D}$.

It is very important to specify that Equation (72(72) $\begin{array}{r}E(x,w,w_{(1)},w_{(2)},\dots ,w_{(n)})=0,\end{array}$(72) ) will be strictly self-adjoint, quasi-self-adjoint, nonlinearly self-adjoint, and weak self-adjoint with some specified conditions and Ibragimov [33–35] gave the idea of first three and Gandarias [36] gives the idea of last one. Here we define the main theorem for the conservation laws.

Theorem 5.2

Let us suppose Lie point, Lie-Backlund, or nonlocal symmetry of (72(72) $\begin{array}{r}E(x,w,w_{(1)},w_{(2)},\dots ,w_{(n)})=0,\end{array}$(72) ) is of the form (81) $\begin{array}{r}\mathcal{N}={\psi }^i\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^i}+{\eta }^i\frac{\mathrm{\partial }}{\mathrm{\partial }{w}^i},\end{array}$(81) with a formal Lagrangian L. Conserved vectors for Equations (72(72) $\begin{array}{r}E(x,w,w_{(1)},w_{(2)},\dots ,w_{(n)})=0,\end{array}$(72) ) and (73(73) $\begin{array}{r}{E}^{\star }(x,w,v,w_{(1)},v_{(1)},w_{(2)},v_{(2)}\cdots ,w_{(n)},v_{(n)})=0,\end{array}$(73) ) can be defined as (82) $\begin{array}{rl}{\mathrm{\Theta }}^{x^i}& ={\psi }^i\mathcal{L}+\mathcal{W}[\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{w}_i}-D_j\left(\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{w}_{ij}}\right)\\ & +D_j{D}_k\left(\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{w}_{ijk}}\right)+\cdots ]\\ & +D_j(\mathcal{W})[\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{w}_{ij}}-D_k\left(\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{w}_{ijk}}\right)+\cdots ]\\ & +D_j{D}_k(\mathcal{W})[\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{w}_{ijk}}+\cdots ],\end{array}$(82) where $\mathcal{W}$ is named as the Lie characteristic function and can be obtained from (83) $\begin{array}{r}\mathcal{W}=\eta -{\psi }^i{w}_i,\end{array}$(83) while $D_i({\mathrm{\Theta }}^{x^i})=0$ is a conservation law.

5.1. Self-adjoint classification for Equation (eqn3)

This section describes the nonlinear self-adjointness classification of Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ). Assume the formal Lagrangian $\mathcal{L}$ as (84) $\begin{array}{r}\mathcal{L}=v[w_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}].\end{array}$(84) Equation (73(73) $\begin{array}{r}{E}^{\star }(x,w,v,w_{(1)},v_{(1)},w_{(2)},v_{(2)}\cdots ,w_{(n)},v_{(n)})=0,\end{array}$(73) ) yields (85) $\begin{array}{r}{E}^{\star }\equiv \frac{\delta }{\delta w}[v(w_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx})]=0,\end{array}$(85) which takes the following form (86) $\begin{array}{r}{E}^{\star }=v_t+\alpha w{v}_x+\beta w^2{v}_x+\gamma v_{xxx}=0.\end{array}$(86) We want an explicit expression of $F(x,t,u)\ne 0$ for Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) that satisfies Equation (86(86) $\begin{array}{r}{E}^{\star }=v_t+\alpha w{v}_x+\beta w^2{v}_x+\gamma v_{xxx}=0.\end{array}$(86) ). (87) $\begin{array}{r}{E}^{\star }{|}_{v=F(x,w)}=\mu E.\end{array}$(87) If we choose $v=F(x,w)$, then we arrive at the self-adjoint condition and then follow different monomials as the set $\{1,w,w_t,w_x,w{w}_x,w^2,w^2{w}_x,w_x^2,w_{xx}\}$ is linearly independent so we come to the condition (88) $\begin{array}{r}{F}_u=\mu ,F_x=0,F_t=0.\end{array}$(88) We get the following solution of Equation (88(88) $\begin{array}{r}{F}_u=\mu ,F_x=0,F_t=0.\end{array}$(88) ) (89) $\begin{array}{r}F=C_{1}w+C_2.\end{array}$(89) This result leads to the following theorem.

Theorem 5.3

Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) is not weak self-adjoint, quasi-self-adjoint, or strictly self-adjoint. After all, Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) is nonlinearly self-adjoint for v = F where F follows from Equation (89(89) $\begin{array}{r}F=C_{1}w+C_2.\end{array}$(89) ).

5.2. Local conservation laws of Equation (eqn3)

The adjoint equation and the symmetries of Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) are now used to establish the local conservation laws. The adjoint equation for Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) is provided by (90) $\begin{array}{r}{E}^{\star }=v_t+\alpha w{v}_x+\beta w^2{v}_x+\gamma v_{xxx}=0.\end{array}$(90) Moreover, the symmetrized version of the Lagrangian is (91) $\begin{array}{r}\mathcal{L}=(C_{1}w+C_2)[w_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}].\end{array}$(91) The three-dimensional Lie algebra for Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ) is given by Equation (10(10) $\begin{array}{r}\begin{array}{rl}{\mathrm{\aleph }}_1& =\frac{\mathrm{\partial }}{\mathrm{\partial }t},\\ {\mathrm{\aleph }}_2& =\frac{\mathrm{\partial }}{\mathrm{\partial }x},\\ {\mathrm{\aleph }}_3& =({\alpha }^{2}t-2\beta x)\frac{\mathrm{\partial }}{\mathrm{\partial }x}-6\beta t\frac{\mathrm{\partial }}{\mathrm{\partial }t}+(2\beta w+\alpha )\frac{\mathrm{\partial }}{\mathrm{\partial }w}.\end{array}\end{array}$(10) ).

For the flux $F_1=w(w_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx})$, we have the conserved vectors as follows;

1. For the symmetry generator ${\mathrm{\aleph }}_1=\frac{\mathrm{\partial }}{\mathrm{\partial }t}$, we have the conserved densities $\begin{array}{rl}{\mathrm{\Theta }}_1^t& =\alpha w^2{w}_x+\beta w^3{w}_x+\gamma w{w}_{xxx},\\ {\mathrm{\Theta }}_1^x& =-\alpha w^2{w}_t-\beta w^3{w}_t-\gamma w_t{w}_{xx}\\ & +\gamma w_x{w}_{tx}-\gamma w{w}_{txx}.\end{array}$

2. For the symmetry generator ${\mathrm{\aleph }}_2=\frac{\mathrm{\partial }}{\mathrm{\partial }x}$, we have the conserved densities $\begin{array}{rl}{\mathrm{\Theta }}_2^t& =-w{w}_x,\\ {\mathrm{\Theta }}_2^x& =w{w}_t.\end{array}$

3. For the symmetry generator ${\mathrm{\aleph }}_3=({\alpha }^{2}t-2\beta x)\frac{\mathrm{\partial }}{\mathrm{\partial }x}-6\beta t\frac{\mathrm{\partial }}{\mathrm{\partial }t}+(2\beta w+\alpha )\frac{\mathrm{\partial }}{\mathrm{\partial }w}$, we have the conserved densities $\begin{array}{rl}{\mathrm{\Theta }}_3^t& =-6\alpha \beta t{w}^2{w}_x-6{\beta }^{2}t{w}^3{w}_x-6\beta \gamma tw{w}_{xxx}\\ & +2\beta w^2+\alpha w-{\alpha }^{2}tw{w}_x+2\beta xw{w}_x,\\ {\mathrm{\Theta }}_3^x& ={\alpha }^{2}tw{w}_t+{\alpha }^2\beta t{w}^3{w}_x-2\beta xw{w}_t+2\alpha \beta w^3\\ & +{\alpha }^2{w}^2-{\alpha }^2\beta t{w}^3+6{\beta }^{2}t{w}^3{w}_t\\ & +8\beta \gamma w{w}_{xx}+\alpha \gamma w{w}_x+6\beta \gamma t{w}_t{w}_{xx}\\ & -4\beta \gamma w_x^2-6\beta \gamma t{w}_x{w}_{tx}+6\beta \gamma tw{w}_{txx}.\end{array}$

For the flux $F_2=w_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}$, we have the conserved vectors as follows;

1. For the symmetry generator ${\mathrm{\aleph }}_1=\frac{\mathrm{\partial }}{\mathrm{\partial }t}$, we have the conserved densities $\begin{array}{rl}{\mathrm{\Theta }}_4^t& =\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx},\\ {\mathrm{\Theta }}_4^x& =-\alpha w{w}_t-\gamma w{w}_{txx}-\beta w^2{w}_t.\end{array}$

2. For the symmetry generator ${\mathrm{\aleph }}_2=\frac{\mathrm{\partial }}{\mathrm{\partial }x}$, we have the conserved densities $\begin{array}{rl}{\mathrm{\Theta }}_5^t& =-w_x,\\ {\mathrm{\Theta }}_5^x& =w_t.\end{array}$

3. For the symmetry generator ${\mathrm{\aleph }}_3=({\alpha }^{2}t-2\beta x)\frac{\mathrm{\partial }}{\mathrm{\partial }x}-6\beta t\frac{\mathrm{\partial }}{\mathrm{\partial }t}+(2\beta w+\alpha )\frac{\mathrm{\partial }}{\mathrm{\partial }w}$, we have the conserved densities $\begin{array}{rl}{\mathrm{\Theta }}_6^t& =-6\alpha \beta tw{w}_x-6{\beta }^{2}t{w}^2{w}_x-6\beta \gamma t{w}_{xxx}\\ & +2\beta w+\alpha -{\alpha }^{2}t{w}_x+2\beta x{w}_x.\\ {\mathrm{\Theta }}_6^x& ={\alpha }^{2}t{w}_t+{\alpha }^{3}tw{w}_x-2\beta x{w}_t+2\alpha \beta w\\ & +{\alpha }^{2}w-{\alpha }^{2}tw{w}_x+6\alpha \beta tw{w}_t+2{\beta }^2{w}^3\\ & +\alpha \beta w^2+6{\beta }^{2}t{w}^2{w}_t+6\beta \gamma w_{xx}+6\beta \gamma t{w}_{txx}.\end{array}$We note that the conservation laws listed above follow the relation $D_i({\mathrm{\Theta }}^{x^i})=0$.

6. Graphical illustrations

The graphical aspects of the physical models help us understand the nature of the model. The solution surface of a model indicates the actual phenomenon which is why the graphical aspects are of more interest. Here we elaborate on the wave nature of some obtained solutions. We plot the solutions $w_{16}$, $w_{20}$, $w_{24}$ and $w_{28}$ to explain the graphical aspects of the nonlinear ESE Equation (3(3) $\begin{array}{r}{w}_t+\alpha w{w}_x+\beta w^2{w}_x+\gamma w_{xxx}=0,\end{array}$(3) ). In Figure 1, you can observe the singular periodic solutions, while Figure 2 illustrates the singular solution. Figure 3 showcases an optical dark soliton solution, and Figure 4 portrays a singular soliton solution. Two-dimensional plots at different time steps are plotted. Each plot is drawn with a particular set of parameters. Density plots via Wolfram Mathematica are also included to indicate the behaviour of the variables in the geometric region.

Figure 1. Graphical aspects of singular periodic solution $w_{16}$ by setting all parameters to unity. (a) 3D plot, (b) 2D plot, (c) density plot and (d) 3D plot.

Figure 2. Graphical aspects of singular solution $w_{20}$ by setting all parameters to unity except ${\mathrm{\Psi }}_2=2$. (a) 3D plot, (b) 2D plot, (c) density plot and (d) 3D plot.

Figure 3. Graphical aspects of optical dark soliton solution $w_{28}$ by setting all parameters to unity except ${\mathrm{\Psi }}_3=-1$. (a) 3D plot, (b) 2D plot, (c) density plot and (d) 3D plot.

Figure 4. Graphical aspects of singular soliton solution $w_{29}$ by setting all parameters to unity except ${\mathrm{\Psi }}_3=-1$. (a) 3D plot, (b) 2D plot, (c) density plot and (d) 3D plot.

7. Concluding remarks

For the nonlinear ESE Equation, the Lie group method was successfully and profitably used to get several periodic wave solutions and numerous soliton solutions. These findings demonstrated that the ESE Equation allows a large number of accurate closed-form solutions with three freely chosen parameters. These free parameters in the derived solutions are determined within the context of deriving the precise solitary wave solutions and several soliton solutions. A variety of nonlinear complicated physical phenomena were manifest in the obtained solutions. In Figures 1–4, the representations corresponded to singular periodic solutions, a singular solution, an optical dark soliton solution, and a singular soliton solution. Our obtained solutions are new and novel for the theory of nonlinear ESE Equation. The solitary waves theory and nonlinear wave phenomena benefit from and use these solutions. It is important to note that the implemented approach was created as an effective, promising, and simple mathematical instrument to deal with any NEEs appearing in nonlinear sciences, plasma physics, mathematical physics, fibre optics, and areas of engineering by applying the Lie group method. In our opinion, the many soliton solutions that were discovered should be useful in explaining the complex physical structures that the soliton theory predicts. Through this study, we are motivated to deal with some other NEEs with the same methodology.

Disclosure statement

No potential conflict of interest was reported by the author(s).