Simulation of new waves in applied sciences via Schrödinger equations

Abstract

The perturbed chiral nonlinear Schrö dinger equation (PCNLSE) reflects the quantum actions such as quantum pictures of Bohm potential and internal self-potential properties. Indeed, this equation introduces the basics of the hidden variable theory in quantum mechanics. Two unified solver methods and exp$(-\phi (\xi \left)\right)$-expansion technique applied to PCNLSE to present many solitonic solutions in an explicit and effective way. The behaviour of these solutions is of qualitatively different structural natures, relying on physical coefficient parameters. The application of three mathematical techniques to our model system provides us with several possible physical property solutions that account for the majority of many phenomena the model under study attempts to depict. The reported bright explosive envelopes, explosive solitons, periodic blow up, bright periodic envelope and huge solitary waves are highly applicable in plasma and nuclear physics, optical communications, electro-magnetic propagations, superfluid and in a lot other applied sciences. The results of this system's solitary structures are consistent with the characteristics of the nonlinear Schrödinger equation systems used to study dispersive modes and higher-order perturbed systems. For more details about the physical dynamical representation of the presented solutions, we have illustrated them with profile pictures using Mathematica and Matlab 18, to obtain complete configurations. The proposed approach can be applied to several equations arising in all applied sciences.

Keywords:

• Perturbative chiral NLSE
• solvers and exp$(-\phi (\xi \left)\right)$-expansion methods
• chiral solitonic structures
• physical applications
• Mathematica and Matlab

AMS subject classifications::

• 34A34
• 35A20
• 35C08
• 35Q51
• 35Q55
• 35Q80

1. Introduction

Investigation of optical solitons has distinctly acquired momentum in the field of the solitary waves through plasma physics [1, 2], bio-mathematics [3], optical fibre [4] and fluid dynamics [5]. Abdelwahed investigated the electron distribution effects on the super periodic wave features for critical plasma densities described by MKP equation [1, 2]. Nonlinear structure characteristics for differential equation systems affected by many factors such as fractional, power law nonlinearity and dissipations which control the solitary pictures in these systems [3–5]. Seismic wave attenuation describes the energy losses experienced by the wave as it travels through the subsurface [6–8]. In fibres of dielectric properties, the efficacious balancing between the effects of optical Kerr and dispersions generates optical soliton. Solitons have indeed drawn a lot of attention due to their durable nature and effective uses in optical communications and long-distance transmissions [9–13]. Due to NLSE's rich physical features, optical solitons, namely, complex and hyperbolic solitons, are now the most interesting field of nonlinear wave propagation in deep water, quantum mechanics, super conductivity, electro-magnetic wave propagation, nuclear physics, magneto-static spin waves and optical fibre communications [14–22]. Various physicists and mathematicians proposed and developed significant standing wave stabilities for NLSEs and related forms such as perturbed, Hartree, unstable, resonant and Choquard forms [23–27]. There are recent developments in analytical and numerical techniques for mathematical models [28–30].

The chiral soliton represents a crucial function in the development of quantum mechanics, namely in Hall effect and neuroscience, where chiral solitons are present. The CNLSE was considered for obtaining bright (dark) soltons [31]. Subsequently in 2004, Bohm potentials (BP) presented in various works were measured with new perturbed NLSE where BP was considered as the perturbative term [32, 33], given by (1) $i{\psi }_t+a{\psi }_{xx}+ib(\psi {\psi }_x^{\ast }-{\psi }^{\ast }{\psi }_x)\psi =i\alpha \psi \frac{\mid \psi {\mid }_{xx}}{\mid \psi \mid },$(1) where ψ and ${\psi }^{\ast }$ are the complex function and its conjugate, $i=\sqrt{-1}$, respectively. a and b denote the coefficients of dispersion and nonlinearity, respectively. The right side denotes the perturbation term, where the coefficient α is BP. Applications of PCNLSE to the advancement of quantum mechanics are outstanding, thus it is so significant to find solutions to PCNLSE for the evolution of quantum mechanics. Various approaches have been applied to extract solutions of the PCNLSE, such as the extended fan sub-equation approach [34], extended modified auxiliary equation mapping approach [35] and many other techniques [36–39].

Our work is driven by the desire to find novel approaches to the PCNLSE solutions, using the two unified solver methods [40–43] and exp$(-\phi (\xi \left)\right)$-expansion technique [44, 45]. The main advantage of utilizing these techniques is to get different families of solutions with free physical parameters. These solutions will be extremely beneficial in many applications such as quantum and nuclear physics, fluid mechanics, plasma physics, nonlinear optics and nanotechnology. Indeed, the proposed solutions are used to study the adiabatic parameter dynamics. The proposed methods are straightforward, sturdy, appropriate and efficient. So, in Section 2 solutions, using the two unified solver techniques and exp$(-\phi (\xi \left)\right)$ method, are presented. Section 3 introduces some physical gained results. Finally, conclusions are given in Section 4.

2. Mathematical analysis

We apply the two unified solver methods [40, 43] and exp$(-\phi (\xi \left)\right)$-expansion technique [44, 45] to extract a variety of chiral solitons for the PCNLSE, utilizing the wave transformation of the following form [34, 35]: (2) $\psi (x,t)=e^{i(-kx+wt+\theta )}\mathrm{\Psi }\left(\zeta \right),\zeta =x+vt,$(2) k is the wave number, w is the frequency of solitons, while θ is the phase constant. Then the following the steps given in [34] give (3) ${\mathrm{\Gamma }}_1\frac{{\mathrm{\partial }}^2\mathrm{\Psi }\left(\zeta \right)}{\mathrm{\partial }{\zeta }^2}+{\mathrm{\Gamma }}_2{\mathrm{\Psi }}^3\left(\zeta \right)+{\mathrm{\Gamma }}_3\mathrm{\Psi }\left(\zeta \right)=0,$(3) where ${\mathrm{\Gamma }}_1=a,{\mathrm{\Gamma }}_2=-2kb,{\mathrm{\Gamma }}_3=-(w+k^{2}a)$.

For complex equation, we have $(v-2ak)\frac{\mathrm{\partial }\mathrm{\Psi }\left(\zeta \right)}{\mathrm{\partial }\zeta }-\alpha \frac{{\mathrm{\partial }}^2\mathrm{\Psi }\left(\zeta \right)}{\mathrm{\partial }{\zeta }^2}=0,$with physical constraints $\begin{array}{rl}& -{\mathrm{\Lambda }}_1{e}^{\frac{\xi \sqrt{(v-2ak{)}^2}}{\alpha }}(\phantom{e^{\frac{\xi (\sqrt{(v-2ak{)}^2}+v)}{\alpha }}}{e}^{\frac{2ak\xi }{\alpha }}(-4a^3{k}^2\\ & -2a^{2}k(\sqrt{(v-2ak{)}^2}-2v)+2{\alpha }^2\omega \\ & +a(v(\sqrt{(v-2ak{)}^2}-v)+2{\alpha }^2{k}^2))\\ & +12{\alpha }^{2}bk{\mathrm{\Lambda }}_2^2{e}^{\frac{\xi (\sqrt{(v-2ak{)}^2}+v)}{\alpha }}\phantom{\frac{2ak\xi }{\alpha }})\\ & +{\mathrm{\Lambda }}_2\left(e^{\frac{2\xi (\sqrt{(v-2ak{)}^2}+ak)}{\alpha }}\right)\\ & \times (\phantom{e^{\frac{\xi (\sqrt{(v-2ak{)}^2}+v)}{\alpha }}}(4a^3{k}^2-2a^{2}k(\sqrt{(v-2ak{)}^2}+2v)-2{\alpha }^2\omega \\ & +a(v(\sqrt{(v-2ak{)}^2}+v)-2{\alpha }^2{k}^2))\\ & -4{\alpha }^{2}bk{\mathrm{\Lambda }}_2^2{e}^{\frac{\xi (3\sqrt{(v-2ak{)}^2}+v)}{\alpha }})\\ & -12{\alpha }^{2}bk{\mathrm{\Lambda }}_2{\mathrm{\Lambda }}_1^2{e}^{\frac{\xi (\sqrt{(v-2ak{)}^2}+v)}{\alpha }}\\ & -4{\alpha }^{2}bk{\mathrm{\Lambda }}_1^3{e}^{\frac{\xi v}{\alpha }}=0,\end{array}$where ${\mathrm{\Lambda }}_1,{\mathrm{\Lambda }}_2$ are constants. Equation (3(3) ${\mathrm{\Gamma }}_1\frac{{\mathrm{\partial }}^2\mathrm{\Psi }\left(\zeta \right)}{\mathrm{\partial }{\zeta }^2}+{\mathrm{\Gamma }}_2{\mathrm{\Psi }}^3\left(\zeta \right)+{\mathrm{\Gamma }}_3\mathrm{\Psi }\left(\zeta \right)=0,$(3) ) represents the dynamical system equation with kinetic energy $K.E=(\frac{1}{2}{k}^2+\frac{1}{2a}\omega ){\mathrm{\Psi }}^2\left(\zeta \right))+\frac{1}{2a}bk{\mathrm{\Psi }}^4\left(\xi \right).$

2.1. Solutions of Equation (1) through solver technique I

The solutions of Equation (3(3) ${\mathrm{\Gamma }}_1\frac{{\mathrm{\partial }}^2\mathrm{\Psi }\left(\zeta \right)}{\mathrm{\partial }{\zeta }^2}+{\mathrm{\Gamma }}_2{\mathrm{\Psi }}^3\left(\zeta \right)+{\mathrm{\Gamma }}_3\mathrm{\Psi }\left(\zeta \right)=0,$(3) ) based on the unified solver approach [40] are as follows:

Rational function solutions: (4) ${\mathrm{\Psi }}_{1,2}(x,t)={(\mp \sqrt{\frac{kb}{a}}(x+vt+\varrho ))}^{-1}.$(4) Consequently, the solutions of Equation (1(1) $i{\psi }_t+a{\psi }_{xx}+ib(\psi {\psi }_x^{\ast }-{\psi }^{\ast }{\psi }_x)\psi =i\alpha \psi \frac{\mid \psi {\mid }_{xx}}{\mid \psi \mid },$(1) ) given by (5) ${\psi }_{1,2}(x,t)=e^{i(-kx+wt+\theta )}{(\mp \sqrt{\frac{kb}{a}}(x+vt+\varrho ))}^{-1}.$(5) Trigonometric function solutions: (6) $\begin{array}{rl}{\mathrm{\Psi }}_{3,4}(x,t)& =\pm \sqrt{\frac{w+k^{2}a}{2kb}}\\ & \times tan\left(\sqrt{\frac{w+k^{2}a}{2a}}(x+vt+\varrho )\right)\end{array}$(6) and (7) $\begin{array}{rl}{\mathrm{\Psi }}_{5,6}(x,t)& =\pm \sqrt{\frac{w+k^{2}a}{2kb}}\\ & \times cot\left(\sqrt{\frac{w+k^{2}a}{2a}}(x+vt+\varrho )\right).\end{array}$(7) Consequently, the solutions of Equation (1(1) $i{\psi }_t+a{\psi }_{xx}+ib(\psi {\psi }_x^{\ast }-{\psi }^{\ast }{\psi }_x)\psi =i\alpha \psi \frac{\mid \psi {\mid }_{xx}}{\mid \psi \mid },$(1) ) given by (8) $\begin{array}{rl}{\psi }_{3,4}(x,t)& =\pm \sqrt{\frac{w+k^{2}a}{2kb}}{e}^{i(-kx+wt+\theta )}\\ & \times tan\left(\sqrt{\frac{w+k^{2}a}{2a}}(x+vt+\varrho )\right)\end{array}$(8) and (9) $\begin{array}{rl}{\psi }_{5,6}(x,t)& =\pm \sqrt{\frac{w+k^{2}a}{2kb}}{e}^{i(-kx+wt+\theta )}\\ & \times cot\left(\sqrt{\frac{w+k^{2}a}{2a}}(x+vt+\varrho )\right).\end{array}$(9) Hyperbolic function solutions: (10) $\begin{array}{rl}{\mathrm{\Psi }}_{7,8}(x,t)& =\pm \sqrt{\frac{-(w+k^{2}a)}{2kb}}\\ & \times tanh\left(\sqrt{\frac{-(w+k^{2}a)}{2a}}(x+vt+\varrho )\right)\end{array}$(10) and (11) $\begin{array}{rl}{\mathrm{\Psi }}_{9,10}(x,t)& =\pm \sqrt{\frac{-(w+k^{2}a)}{2kb}}\\ & \times coth\left(\sqrt{\frac{-(w+k^{2}a)}{2a}}(x+vt+\varrho )\right).\end{array}$(11) Consequently, the solutions of Equation (1(1) $i{\psi }_t+a{\psi }_{xx}+ib(\psi {\psi }_x^{\ast }-{\psi }^{\ast }{\psi }_x)\psi =i\alpha \psi \frac{\mid \psi {\mid }_{xx}}{\mid \psi \mid },$(1) ) are given by (12) $\begin{array}{rl}{\psi }_{7,8}(x,t)& =\pm \sqrt{\frac{-(w+k^{2}a)}{2kb}}{e}^{i(-kx+wt+\theta )}\\ & \times tanh\left(\sqrt{\frac{-(w+k^{2}a)}{2a}}(x+vt+\varrho )\right)\end{array}$(12) and (13) $\begin{array}{rl}{\psi }_{9,10}(x,t)& =\pm \sqrt{\frac{-(w+k^{2}a)}{2kb}}{e}^{i(-kx+wt+\theta )}\\ & \times coth\left(\sqrt{\frac{-(w+k^{2}a)}{2a}}(x+vt+\varrho )\right).\end{array}$(13) Here ϱ is an arbitrary constant.

2.2. Solutions of Equation (1) through solver technique II

The solutions of Equation (3(3) ${\mathrm{\Gamma }}_1\frac{{\mathrm{\partial }}^2\mathrm{\Psi }\left(\zeta \right)}{\mathrm{\partial }{\zeta }^2}+{\mathrm{\Gamma }}_2{\mathrm{\Psi }}^3\left(\zeta \right)+{\mathrm{\Gamma }}_3\mathrm{\Psi }\left(\zeta \right)=0,$(3) ) based on the unified solver approach [43] are as follows: (14) $\begin{array}{rl}{\hat{\mathrm{\Psi }}}_{1,2}(x,t)& =\pm \sqrt{\frac{(w+k^{2}a)}{-kb}}\\ & \times sech\left(\pm \sqrt{\frac{w+k^{2}a}{a}}(x+vt)\right),\end{array}$(14) and (15) $\begin{array}{rl}{\hat{\mathrm{\Psi }}}_{3,4}(x,t)& =\pm \sqrt{\frac{35(w+k^{2}a)}{-36kb}}\\ & \times sec{h}^2(\pm \sqrt{\frac{5(w+k^{2}a)}{12a}}(x+vt\left)\right).\end{array}$(15) Consequently, the solutions of Equation (1(1) $i{\psi }_t+a{\psi }_{xx}+ib(\psi {\psi }_x^{\ast }-{\psi }^{\ast }{\psi }_x)\psi =i\alpha \psi \frac{\mid \psi {\mid }_{xx}}{\mid \psi \mid },$(1) ) are given by (16) $\begin{array}{rl}{\hat{\psi }}_{1,2}(x,t)& =\pm \sqrt{\frac{(w+k^{2}a)}{-kb}}{e}^{i(-kx+wt+\theta )}\\ & \times sech\left(\pm \sqrt{\frac{w+k^{2}a}{a}}(x+vt)\right)\end{array}$(16) and (17) $\begin{array}{rl}{\hat{\psi }}_{3,4}(x,t)& =\pm \sqrt{\frac{35(w+k^{2}a)}{-36kb}}{e}^{i(-kx+wt+\theta )}\\ & \times sec{h}^2(\pm \sqrt{\frac{5(w+k^{2}a)}{12a}}(x+vt\left)\right).\end{array}$(17)

2.3. Solutions of Equation (1) through the exp$(-\phi (\xi \left)\right)$-expansion approach

In view of exp$(-\phi (\xi \left)\right)$-expansion approach [44, 45], the solutions of Equation (3(3) ${\mathrm{\Gamma }}_1\frac{{\mathrm{\partial }}^2\mathrm{\Psi }\left(\zeta \right)}{\mathrm{\partial }{\zeta }^2}+{\mathrm{\Gamma }}_2{\mathrm{\Psi }}^3\left(\zeta \right)+{\mathrm{\Gamma }}_3\mathrm{\Psi }\left(\zeta \right)=0,$(3) ) are (18) $\begin{array}{r}\mathrm{\Psi }=A_0+A_{1}exp(-\phi (\zeta \left)\right),{\phi }^{\mathrm{\prime }-\phi \left(\zeta \right)}+\mu e^{\phi \left(\zeta \right)}+\lambda ,\end{array}$(18) $A_0$ and $A_1\ne 0$ are constants. Superseding Ψ, ${\mathrm{\Psi }}^{\mathrm{\prime }\mathrm{\prime }}$, ${\mathrm{\Psi }}^3$ into Equation (3(3) ${\mathrm{\Gamma }}_1\frac{{\mathrm{\partial }}^2\mathrm{\Psi }\left(\zeta \right)}{\mathrm{\partial }{\zeta }^2}+{\mathrm{\Gamma }}_2{\mathrm{\Psi }}^3\left(\zeta \right)+{\mathrm{\Gamma }}_3\mathrm{\Psi }\left(\zeta \right)=0,$(3) ) and putting the coefficients of $exp(-\phi )$ to zero produce a system of algebraic equations. Solving these equations yields the following: (19) $\begin{array}{r}u\left(\xi \right)=\pm \frac{\sqrt{a}}{2\sqrt{kb}}(\lambda +2exp(-\phi (\zeta \left)\right)).\end{array}$(19) Then the solutions of Equation (3(3) ${\mathrm{\Gamma }}_1\frac{{\mathrm{\partial }}^2\mathrm{\Psi }\left(\zeta \right)}{\mathrm{\partial }{\zeta }^2}+{\mathrm{\Gamma }}_2{\mathrm{\Psi }}^3\left(\zeta \right)+{\mathrm{\Gamma }}_3\mathrm{\Psi }\left(\zeta \right)=0,$(3) ) are

Case 1. For $\mu \ne 0,L>0,$ (20) $\begin{array}{rl}{\tilde{\mathrm{\Psi }}}_{1,2}(x,t)& =\pm \frac{\sqrt{a}\lambda }{2\sqrt{kb}}\\ & \times (\lambda -\frac{4\mu }{\sqrt{L}tanh\left(\frac{\sqrt{L}}{2}(x+vt+C)\right)+\lambda }),\end{array}$(20) (21) $\begin{array}{rl}{\tilde{\mathrm{\Psi }}}_{3,4}(x,t)& =\pm \frac{\sqrt{a}\lambda }{2\sqrt{kb}}\\ & \times (\lambda -\frac{4\mu }{\sqrt{L}coth\left(\frac{\sqrt{L}}{2}(x+vt+C)\right)+\lambda }).\end{array}$(21) Consequently, the solutions of Equation (1(1) $i{\psi }_t+a{\psi }_{xx}+ib(\psi {\psi }_x^{\ast }-{\psi }^{\ast }{\psi }_x)\psi =i\alpha \psi \frac{\mid \psi {\mid }_{xx}}{\mid \psi \mid },$(1) ) are given by (22) $\begin{array}{rl}{\tilde{\psi }}_{1,2}(x,t)& =\pm \frac{\sqrt{a}\lambda }{2\sqrt{kb}}{e}^{i(-kx+wt+\theta )}\\ & \times (\lambda -\frac{4\mu }{\sqrt{L}tanh\left(\frac{\sqrt{L}}{2}(x+vt+C)\right)+\lambda }).\end{array}$(22) (23) $\begin{array}{rl}{\tilde{\psi }}_{3,4}(x,t)& =\pm \frac{\sqrt{a}\lambda }{2\sqrt{kb}}{e}^{i(-kx+wt+\theta )}\\ & \times (\lambda -\frac{4\mu }{\sqrt{L}coth\left(\frac{\sqrt{L}}{2}(x+vt+C)\right)+\lambda }).\end{array}$(23) Case 2. For $\mu \ne 0,L<0,$ (24) $\begin{array}{rl}& {\tilde{\mathrm{\Psi }}}_{5,6}(x,t)=\pm \frac{\sqrt{a}}{2\sqrt{kb}}\\ & \times (\lambda +\frac{4\mu }{\sqrt{-L}tan\left(\frac{\sqrt{-L}}{2}(x+vt+C)\right)-\lambda }).\end{array}$(24) (25) $\begin{array}{rl}& {\tilde{\mathrm{\Psi }}}_{7,8}(x,t)=\pm \frac{\sqrt{a}}{2\sqrt{kb}}\\ & \times (\lambda +\frac{4\mu }{\sqrt{-L}cot\left(\frac{\sqrt{-L}}{2}(x+vt+C)\right)-\lambda }).\end{array}$(25) Consequently, the solutions of Equation (1(1) $i{\psi }_t+a{\psi }_{xx}+ib(\psi {\psi }_x^{\ast }-{\psi }^{\ast }{\psi }_x)\psi =i\alpha \psi \frac{\mid \psi {\mid }_{xx}}{\mid \psi \mid },$(1) ) are given by (26) $\begin{array}{rl}& {\tilde{\psi }}_{5,6}(x,t)=\pm \frac{\sqrt{a}}{2\sqrt{kb}}{e}^{i(-kx+wt+\theta )}\\ & \times (\lambda +\frac{4\mu }{\sqrt{-L}tan\left(\frac{\sqrt{-L}}{2}(x+vt+C)\right)-\lambda }).\end{array}$(26) (27) $\begin{array}{rl}& {\tilde{\psi }}_{7,8}(x,t)=\pm \frac{\sqrt{a}}{2\sqrt{kb}}{e}^{i(-kx+wt+\theta )}\\ & \times (\lambda +\frac{4\mu }{\sqrt{-L}cot\left(\frac{\sqrt{-L}}{2}(x+vt+C)\right)-\lambda }).\end{array}$(27) Case 3. For $\mu =0,\lambda \ne 0,L>0,$ (28) $\begin{array}{r}{\tilde{\mathrm{\Psi }}}_{9,10}(x,t)=\pm \frac{\sqrt{a}}{2\sqrt{kb}}(\lambda +\frac{2\lambda }{exp\left(\lambda (x+vt+C)\right)-1}).\end{array}$(28) Consequently, the solutions of Equation (1(1) $i{\psi }_t+a{\psi }_{xx}+ib(\psi {\psi }_x^{\ast }-{\psi }^{\ast }{\psi }_x)\psi =i\alpha \psi \frac{\mid \psi {\mid }_{xx}}{\mid \psi \mid },$(1) ) are given by (29) $\begin{array}{rl}{\tilde{\psi }}_{9,10}(x,t)& =\pm \frac{\sqrt{a}}{2\sqrt{kb}}{e}^{i(-kx+wt+\theta )}\\ & \times (\lambda +\frac{2\lambda }{exp\left(\lambda (x+vt+C)\right)-1}).\end{array}$(29) Case 4. For $\mu \ne 0,\lambda \ne 0,L=0,$ (30) $\begin{array}{r}{\tilde{\mathrm{\Psi }}}_{11,12}(x,t)=\pm \frac{\sqrt{a}}{2\sqrt{kb}}(\lambda -\frac{{\lambda }^2(\xi +C)}{\lambda (x+vt+C)+2}).\end{array}$(30) Consequently, the solutions of Equation (1(1) $i{\psi }_t+a{\psi }_{xx}+ib(\psi {\psi }_x^{\ast }-{\psi }^{\ast }{\psi }_x)\psi =i\alpha \psi \frac{\mid \psi {\mid }_{xx}}{\mid \psi \mid },$(1) ) are given by (31) $\begin{array}{rl}{\tilde{\psi }}_{11,12}(x,t)& =\pm \frac{\sqrt{a}}{2\sqrt{kb}}{e}^{i(-kx+wt+\theta )}\\ & \times (\lambda -\frac{{\lambda }^2(\xi +C)}{\lambda (x+vt+C)+2}).\end{array}$(31) Case 5. At $\mu =0,\lambda =0,L=0,$ (32) $\begin{array}{r}{\tilde{\mathrm{\Psi }}}_{13,14}(x,t)=\pm \frac{\sqrt{a}}{2\sqrt{kb}}\left(\frac{1}{x+vt+C}\right).\end{array}$(32) Consequently, the solutions of Equation (1(1) $i{\psi }_t+a{\psi }_{xx}+ib(\psi {\psi }_x^{\ast }-{\psi }^{\ast }{\psi }_x)\psi =i\alpha \psi \frac{\mid \psi {\mid }_{xx}}{\mid \psi \mid },$(1) ) are given by (33) $\begin{array}{r}{\tilde{\psi }}_{13,14}(x,t)=\pm \frac{\sqrt{a}}{2\sqrt{kb}}{e}^{i(-kx+wt+\theta )}\left(\frac{1}{x+vt+C}\right).\end{array}$(33) Here $L={\lambda }^2-4\mu$ and C is constant.

3. Results and discussion

Since the perturbed term in PCNLSE represented the quantum perturbed potential is which known as Bohm potential and introduces the internal self-potentials for quantum properties. So, some new solitary solutions for PCNLSE were derived explicitly by using two solvers and exp$(-\phi (\xi \left)\right)$-expansion techniques. The investigation of PCNLSE performs a crucial way in the evolution of quantum Hall effect, in which chiral excitations are familiar with manifesting [46, 47]. The well-obtained structural behaviours of the propagating solutions for Equation (1(1) $i{\psi }_t+a{\psi }_{xx}+ib(\psi {\psi }_x^{\ast }-{\psi }^{\ast }{\psi }_x)\psi =i\alpha \psi \frac{\mid \psi {\mid }_{xx}}{\mid \psi \mid },$(1) ) being rationals, dark envelopes, periodicals, explosive envelopes, shock-likes, localized solitons, dissipative, localized huge waves, etc., and established on physical parameter contributions in PCNLSE coefficients. For example, at specific wave number values, the wave's attitude changes for critical nodes and stability zones turn into unstable regions [48, 49]. Nonlinear dispersive fibres transmit bright and dark solitons. Waves with lower intensities than the background are identified by dark solitons. However, the bright-type has peak intensities that are greater than the background [50]. Here, we introduce 3D selected solution profiles for PCNLSE via physical parameter variations. Figures 1–4 represent the dynamical behaviours of some selected solutions using the first solver. It was noted that bright explosive envelope and explosive soliton propagations are obtained for solution (5(5) ${\psi }_{1,2}(x,t)=e^{i(-kx+wt+\theta )}{(\mp \sqrt{\frac{kb}{a}}(x+vt+\varrho ))}^{-1}.$(5) ) in Figures 1 and 2. In Figure 1 the wave changes periodically in the x axis and extremely quick rapids in the $xt$ plane. For solution (9(9) $\begin{array}{rl}{\psi }_{5,6}(x,t)& =\pm \sqrt{\frac{w+k^{2}a}{2kb}}{e}^{i(-kx+wt+\theta )}\\ & \times cot\left(\sqrt{\frac{w+k^{2}a}{2a}}(x+vt+\varrho )\right).\end{array}$(9) ) a dissipative and periodic blow-up solitary waves are depicted in Figures 3 and 4. For second solver outputs, solution (16(16) $\begin{array}{rl}{\hat{\psi }}_{1,2}(x,t)& =\pm \sqrt{\frac{(w+k^{2}a)}{-kb}}{e}^{i(-kx+wt+\theta )}\\ & \times sech\left(\pm \sqrt{\frac{w+k^{2}a}{a}}(x+vt)\right)\end{array}$(16) ) represents a bright periodic envelope and stable stationary solitonic structures with constant amplitudes and energies in Figures 5 and 6. Also, the solution (17(17) $\begin{array}{rl}{\hat{\psi }}_{3,4}(x,t)& =\pm \sqrt{\frac{35(w+k^{2}a)}{-36kb}}{e}^{i(-kx+wt+\theta )}\\ & \times sec{h}^2(\pm \sqrt{\frac{5(w+k^{2}a)}{12a}}(x+vt\left)\right).\end{array}$(17) ) denotes a huge and soliton waves, as shown in Figures 7 and 8. It was noted that these wave structures are conservative dynamic waves with amplitudes depending mainly on the model parameters.

Figure 1. Change of real part of solution (5(5) ${\psi }_{1,2}(x,t)=e^{i(-kx+wt+\theta )}{(\mp \sqrt{\frac{kb}{a}}(x+vt+\varrho ))}^{-1}.$(5) ) with x, t.

Figure 2. Change of $\left|{\psi }_1\right(x,t\left)\right|$ in Equation (5(5) ${\psi }_{1,2}(x,t)=e^{i(-kx+wt+\theta )}{(\mp \sqrt{\frac{kb}{a}}(x+vt+\varrho ))}^{-1}.$(5) ) with x, t.

Figure 3. Change of real part of solution (9(9) $\begin{array}{rl}{\psi }_{5,6}(x,t)& =\pm \sqrt{\frac{w+k^{2}a}{2kb}}{e}^{i(-kx+wt+\theta )}\\ & \times cot\left(\sqrt{\frac{w+k^{2}a}{2a}}(x+vt+\varrho )\right).\end{array}$(9) ) with x, t.

Figure 4. Change of $\left|{\psi }_3\right(x,t\left)\right|$ in Equation (9(9) $\begin{array}{rl}{\psi }_{5,6}(x,t)& =\pm \sqrt{\frac{w+k^{2}a}{2kb}}{e}^{i(-kx+wt+\theta )}\\ & \times cot\left(\sqrt{\frac{w+k^{2}a}{2a}}(x+vt+\varrho )\right).\end{array}$(9) ) with x, t.

Figure 5. Change of real part of solution (16(16) $\begin{array}{rl}{\hat{\psi }}_{1,2}(x,t)& =\pm \sqrt{\frac{(w+k^{2}a)}{-kb}}{e}^{i(-kx+wt+\theta )}\\ & \times sech\left(\pm \sqrt{\frac{w+k^{2}a}{a}}(x+vt)\right)\end{array}$(16) ) with x, t.

Figure 6. Change of $\left|{\hat{\psi }}_1\right(x,t\left)\right|$ in Equation (16(16) $\begin{array}{rl}{\hat{\psi }}_{1,2}(x,t)& =\pm \sqrt{\frac{(w+k^{2}a)}{-kb}}{e}^{i(-kx+wt+\theta )}\\ & \times sech\left(\pm \sqrt{\frac{w+k^{2}a}{a}}(x+vt)\right)\end{array}$(16) ) with x, t.

Figure 7. Change of real part of solution (17(17) $\begin{array}{rl}{\hat{\psi }}_{3,4}(x,t)& =\pm \sqrt{\frac{35(w+k^{2}a)}{-36kb}}{e}^{i(-kx+wt+\theta )}\\ & \times sec{h}^2(\pm \sqrt{\frac{5(w+k^{2}a)}{12a}}(x+vt\left)\right).\end{array}$(17) ) with x, t.

Figure 8. Change of $\left|{\hat{\psi }}_3\right(x,t\left)\right|$ in Equation (17(17) $\begin{array}{rl}{\hat{\psi }}_{3,4}(x,t)& =\pm \sqrt{\frac{35(w+k^{2}a)}{-36kb}}{e}^{i(-kx+wt+\theta )}\\ & \times sec{h}^2(\pm \sqrt{\frac{5(w+k^{2}a)}{12a}}(x+vt\left)\right).\end{array}$(17) ) with x, t.

The results of $\exp (-\phi (\xi \left)\right)$-expansion are shown in Figures 9–13. The solution (22(22) $\begin{array}{rl}{\tilde{\psi }}_{1,2}(x,t)& =\pm \frac{\sqrt{a}\lambda }{2\sqrt{kb}}{e}^{i(-kx+wt+\theta )}\\ & \times (\lambda -\frac{4\mu }{\sqrt{L}tanh\left(\frac{\sqrt{L}}{2}(x+vt+C)\right)+\lambda }).\end{array}$(22) ) introduces an important dark type envelope solution and new super solitary profile as shown in Figures 9 and 10. It was remarked for Figure 9 that the wave behaves periodic in the x axis but the critical point depends mainly on time t. The constant amplitude super soliton in Figure 10 introduces a new wave form that constrains soliton energy according to physical parameters. In Figure 11, a shock like solitary formation is produced from the solution (26(26) $\begin{array}{rl}& {\tilde{\psi }}_{5,6}(x,t)=\pm \frac{\sqrt{a}}{2\sqrt{kb}}{e}^{i(-kx+wt+\theta )}\\ & \times (\lambda +\frac{4\mu }{\sqrt{-L}tan\left(\frac{\sqrt{-L}}{2}(x+vt+C)\right)-\lambda }).\end{array}$(26) ). This solution is converted from a solitonic form to a shock dissipative form depending on the parameter values. Finally, bright explosive sharp envelope and explosive solitonic excitations are formed as shown in Figures 12 and 13 from solution (29(29) $\begin{array}{rl}{\tilde{\psi }}_{9,10}(x,t)& =\pm \frac{\sqrt{a}}{2\sqrt{kb}}{e}^{i(-kx+wt+\theta )}\\ & \times (\lambda +\frac{2\lambda }{exp\left(\lambda (x+vt+C)\right)-1}).\end{array}$(29) ). The multiple use of mathematical solutions methods for the same studied the system give us several solutions with different properties that may explain most of the phenomena described by the studied system. Some of the solitary solution results of this system agree with the solution properties of the nonlinear Schrödinger equation system, which describe strong dispersive and higher-order systems [20, 51–53].

Figure 9. Change of the real part of solution (22(22) $\begin{array}{rl}{\tilde{\psi }}_{1,2}(x,t)& =\pm \frac{\sqrt{a}\lambda }{2\sqrt{kb}}{e}^{i(-kx+wt+\theta )}\\ & \times (\lambda -\frac{4\mu }{\sqrt{L}tanh\left(\frac{\sqrt{L}}{2}(x+vt+C)\right)+\lambda }).\end{array}$(22) ) with x, t.

Figure 10. Change of $\left|{\tilde{\psi }}_1\right(x,t\left)\right|$ in Equation (22(22) $\begin{array}{rl}{\tilde{\psi }}_{1,2}(x,t)& =\pm \frac{\sqrt{a}\lambda }{2\sqrt{kb}}{e}^{i(-kx+wt+\theta )}\\ & \times (\lambda -\frac{4\mu }{\sqrt{L}tanh\left(\frac{\sqrt{L}}{2}(x+vt+C)\right)+\lambda }).\end{array}$(22) ) with x, t.

Figure 11. Change of the real part of solution (26(26) $\begin{array}{rl}& {\tilde{\psi }}_{5,6}(x,t)=\pm \frac{\sqrt{a}}{2\sqrt{kb}}{e}^{i(-kx+wt+\theta )}\\ & \times (\lambda +\frac{4\mu }{\sqrt{-L}tan\left(\frac{\sqrt{-L}}{2}(x+vt+C)\right)-\lambda }).\end{array}$(26) ) with x, t.

Figure 12. Change of the real part of solution (29(29) $\begin{array}{rl}{\tilde{\psi }}_{9,10}(x,t)& =\pm \frac{\sqrt{a}}{2\sqrt{kb}}{e}^{i(-kx+wt+\theta )}\\ & \times (\lambda +\frac{2\lambda }{exp\left(\lambda (x+vt+C)\right)-1}).\end{array}$(29) ) with x, t.

Figure 13. Change of $\left|{\tilde{\psi }}_9\right(x,t\left)\right|$ in Equation (29(29) $\begin{array}{rl}{\tilde{\psi }}_{9,10}(x,t)& =\pm \frac{\sqrt{a}}{2\sqrt{kb}}{e}^{i(-kx+wt+\theta )}\\ & \times (\lambda +\frac{2\lambda }{exp\left(\lambda (x+vt+C)\right)-1}).\end{array}$(29) ) with x, t.

In summary, general and several new obtained solutions may be with vital applications in strong dispersive and higher-order systems of fluid physics, femtoseconds pulse, capillary studies, nuclear engineering, optical fibre communications, superfluids and switching techniques. The described findings indicate that the approaches are simple, reliable and effective for a wide range of solutions with particular features that may explain the majority of the phenomena. Indeed, these methods are powerful tools to consider other complicated nonlinear partial differential equations arising in applied sciences and new physics.

4. Conclusions

We have displayed a rich diversity of solution classes to PCNLSE, using solver methods and exp$(-\phi (\xi \left)\right)$-expansion techniques by physical coefficient parameters. The proposed techniques are not only straightforward but also easy and confer new vital outcomes as explosive solitons, bright explosive envelopes, periodic blow up, bright periodic envelope and freak solitary waves. These Chiral soliton behaviours are so paramount in evolutions of quantum mechanics of Hall effect in view of quantum Bohm potential and interior self-potential through quantum characteristics. A clarified graphical investigation for some selected equation types utilizing mathematical programmes that are valuable to comprehend more distinctly about the dynamic kinetics of solitary formations. Finally, the offered three techniques may be exercised for another quantum equations arising in different applicable fields.

On the other hand, the utilization of several mathematical solution techniques for the same investigated system provides us with a sufficient number of solutions with different characteristics that might potentially explain the majority of the phenomena the studied system describes. The nonlinear Schrödinger equation system's solutions qualities, which characterize strong dispersive and higher-order systems and certain of the system's solitary solution results are in agreement.

Disclosure statement

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

Data availability statement

All the data are available within the article.

Additional information

Funding

The authors extend their appreciation to the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia for funding this research work through [project number RI-44-0764].