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Soliton and other solutions to the (1【-逻*辑*与-】nbsp;+【-逻*辑*与-】nbsp;2)-dimensional chiral nonlinear Sc

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K Hosseini,1,, M Mirzazadeh,2,1Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran
2Department of Engineering Sciences, Faculty of Technology and Engineering, East of Guilan, University of Guilan, P.C. 44891-63157 Rudsar-Vajargah, Iran

First author contact: Authors to whom any correspondence should be addressed.
Received:2020-07-04Revised:2020-08-10Accepted:2020-09-10Online:2020-12-01


Abstract
The (1+2)-dimensional chiral nonlinear Schrödinger equation (2D-CNLSE) as a nonlinear evolution equation is considered and studied in a detailed manner. To this end, a complex transform is firstly adopted to arrive at the real and imaginary parts of the model, and then, the modified Jacobi elliptic expansion method is formally utilized to derive soliton and other solutions of the 2D-CNLSE. The exact solutions presented in this paper can be classified as topological and nontopological solitons as well as Jacobi elliptic function solutions.
Keywords: modified Jacobi elliptic expansion method;(1 + 2)-dimensional chiral nonlinear Schrödinger equation;topological and nontopological solitons;Jacobi elliptic function solutions


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K Hosseini, M Mirzazadeh. Soliton and other solutions to the (1+2)-dimensional chiral nonlinear Schrödinger equation. Communications in Theoretical Physics, 2020, 72(12): 125008- doi:10.1088/1572-9494/abb87b

1. Introduction

Nishino et al [1] studied a nonlinear evolution equation known as the (1+1)-dimensional chiral nonlinear Schrödinger equation (1D-CNLSE) in the form$\begin{eqnarray*}{\rm{i}}{u}_{t}+{c}_{1}{u}_{xx}+{\rm{i}}{c}_{2}\left(u{u}_{x}^{* }-{u}^{* }{u}_{x}\right)u=0,\end{eqnarray*}$and obtained its bright and dark solitons. The 1D-CNLSE was established as a one-dimensional reduction of a system describing the edge states of the Fractional Quantum Hall Effect [2]. Thereafter, the (1+2)-dimensional form of the chiral nonlinear Schrödinger equation (2D-CNLSE), namely [3-6]$\begin{eqnarray}\begin{array}{l}{\rm{i}}{u}_{t}+{c}_{1}\left({u}_{xx}+{u}_{yy}\right)+{\rm{i}}\left({c}_{2}\left(u{u}_{x}^{* }-{u}^{* }{u}_{x}\right)\right.\\ \,+\,\left.{c}_{3}\left(u{u}_{y}^{* }-{u}^{* }{u}_{y}\right)\right)u=0,\end{array}\end{eqnarray}$was studied by a series of researchers using different methods. For example, Biswas [3] obtained chiral solitons of the 2D-CNLSE using several soliton ansatz methods. Eslami [4] used the trial solution method to derive solitons and other solutions of the 2D-CNLSE. Raza and Javid [5] derived exact solutions of the 2D-CNLSE through the modified extended direct algebraic method. Raza and Arshed [6] utilized the sine-Gordon expansion method to obtain chiral bright and dark soliton solutions of the 2D-CNLSE.

It should be mentioned that the first and second terms appeared in equation (1) present the evolution term and the dispersion term, respectively. Besides, ${c}_{2}$ and ${c}_{3}$ signify the coefficients of nonlinear coupling terms. Unfortunately, the 2D-CNLSE is not Galilean invariant and does not possess the Painlevé test [3]. Such features of the 2D-CNLSE reveal the importance of extracting solitons and other solutions to it.

The current paper aims to present soliton and other solutions of the 2D-CNLSE using the modified Jacobi elliptic expansion (MJEE) method [7-14]. To highlight the effectiveness of the MJEE method for handling nonlinear evolution equations, a review of its recent applications is provided below. Ma et al [7] used the MJEE method to construct new exact solutions of MKdV and BBM equations. Hosseini et al [8] obtained solitons and Jacobi elliptic function solutions of the complex Ginzburg-Landau equation using the MJEE method. The reader is referred to [15-36].

The organization of this paper is as follows: in section 2, the MJEE method is described in detail. In section 3, soliton and other solutions of the 2D-CNLSE are constructed using the MJEE method. Finally, conclusions are presented in the last section.

2. The MJEE method

The key ideas of the MJEE method are summarized in this section. To start, consider the following nonlinear ordinary differential equation$\begin{eqnarray}P\left(u,u^{\prime} ,u^{\prime\prime} ,\,\mathrm{...}\right)=0,^{\prime} =\displaystyle \frac{{\rm{d}}}{{\rm{d}}{\epsilon }}.\end{eqnarray}$

Suppose that the exact solution of equation (2) can be written as a finite series in the form$\begin{eqnarray}\begin{array}{l}u\left({\epsilon }\right)={a}_{0}+\displaystyle {\sum }_{i=1}^{N}{\left(\displaystyle \frac{J\left({\epsilon }\right)}{1+{J}^{2}\left({\epsilon }\right)}\right)}^{i-1}\left({a}_{i}\displaystyle \frac{J\left({\epsilon }\right)}{1+{J}^{2}\left({\epsilon }\right)}\right.\\ \,\,\,+\,\left.{b}_{i}\displaystyle \frac{1-{J}^{2}\left({\epsilon }\right)}{1+{J}^{2}\left({\epsilon }\right)}\right),\,{a}_{N}\,{\rm{or}}\,{b}_{N}\ne 0,\end{array}\end{eqnarray}$where ${a}_{0},$ ${a}_{i},$ and ${b}_{i}$ ($1\leqslant i\leqslant N$) are determined later, $N$ is obtained by the balance principle, and $J\left({\epsilon }\right)$ is a Jacobi elliptic function satisfying$\begin{eqnarray}{\left(J^{\prime} \left({\epsilon }\right)\right)}^{2}=D+E{J}^{2}\left({\epsilon }\right)+F{J}^{4}\left({\epsilon }\right).\end{eqnarray}$

The exact solutions of the Jacobi elliptic equation (4) depending on the parameters $D,$ $E,$ and $F$ have been listed in table 1.


Table 1.
Table 1.Jacobi elliptic function solutions of equation (4).
No.DEFJ(ξ)
1$1$$-\left({m}^{2}+1\right)$${m}^{2}$${\rm{sn}}\left(\xi \right)$
2$1-{m}^{2}$$2{m}^{2}-1$$-{m}^{2}$${\rm{cn}}\left(\xi \right)$
3${m}^{2}$$-\left({m}^{2}+1\right)$$1$${\rm{ns}}\left(\xi \right)$
4$-{m}^{2}$$2{m}^{2}-1$$1-{m}^{2}$${\rm{nc}}\left(\xi \right)$

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By inserting the finite series (3) into equation (2) and exerting some operations, we get a nonlinear algebraic system whose solution results in exact solutions of equation (2).

Several useful properties of the Jacobi elliptic functions have been given below: ${{\rm{sn}}}^{2}\left(\xi \right)+{{\rm{cn}}}^{2}\left(\xi \right)=1.$
${\rm{sn}}\left(\xi \right)={\rm{sn}}\left(\xi ,m\right)\to \,\tanh \left(\xi \right)$ when $m\to 1.$
${\rm{ns}}\left(\xi \right)={\left({\rm{sn}}\left(\xi ,m\right)\right)}^{-1}\to \,\coth \left(\xi \right)$ when $m\to 1.$


3. The 2D-CNLSE and its soliton and other solutions

The main aim of this section is to present soliton and other solutions of the 2D-CNLSE using the MJEE method. To this end, a complex transformation is firstly considered as$\begin{eqnarray}u\left(x,y,t\right)=U\left({\epsilon }\right){{\rm{e}}}^{{\rm{i}}\left({\kappa }_{2}x+{\lambda }_{2}y+{\mu }_{2}t\right)},{\epsilon }={\kappa }_{1}x+{\lambda }_{1}y-{\mu }_{1}t,\end{eqnarray}$where

${\kappa }_{1}:$ The inverse width of the soliton in the $x$-direction,

${\lambda }_{1}:$ The inverse width of the soliton in the $y$-direction,

${\mu }_{1}:$ The velocity of the soliton,

${\kappa }_{2}:$ The frequency in the $x$-direction,

${\lambda }_{2}:$ The frequency in the $y$-direction,

${\mu }_{2}:$ The soliton frequency.

Considering the complex transformation (5) and the 2D-CNLSE (1) gives$\begin{eqnarray}\begin{array}{l}{c}_{1}\left({{\kappa }_{1}}^{2}+{{\lambda }_{1}}^{2}\right)\displaystyle \frac{{{\rm{d}}}^{2}U\left({\epsilon }\right)}{{\rm{d}}{{\epsilon }}^{2}}-\left({c}_{1}{{\kappa }_{2}}^{2}+{c}_{1}{{\lambda }_{2}}^{2}+{\mu }_{2}\right)U\left({\epsilon }\right)\\ \,+\,2\left({c}_{2}{\kappa }_{2}+{c}_{3}{\lambda }_{2}\right){U}^{3}\left({\epsilon }\right)=0,\end{array}\end{eqnarray}$$\begin{eqnarray}\left(2{c}_{1}{\kappa }_{1}{\kappa }_{2}+2{c}_{1}{\lambda }_{1}{\lambda }_{2}-{\mu }_{1}\right)\displaystyle \frac{{\rm{d}}U\left({\epsilon }\right)}{{\rm{d}}{\epsilon }}=0.\end{eqnarray}$

From equation (7), the soliton velocity is found as$\begin{eqnarray*}{\mu }_{1}=2{c}_{1}\left({\kappa }_{1}{\kappa }_{2}+{\lambda }_{1}{\lambda }_{2}\right).\end{eqnarray*}$

Now, balancing the terms $\tfrac{{{\rm{d}}}^{2}U\left({\epsilon }\right)}{{\rm{d}}{{\epsilon }}^{2}}$ and ${U}^{3}\left({\epsilon }\right)$ appeared in equation (6) results in $N=1.$ Consequently, based on the initial assumption of the MJEE method, the solution of equation (6) can be written as follows$\begin{eqnarray}U\left({\epsilon }\right)={a}_{0}+{a}_{1}\displaystyle \frac{J\left({\epsilon }\right)}{1+{J}^{2}\left({\epsilon }\right)}+{a}_{2}\displaystyle \frac{1-{J}^{2}\left({\epsilon }\right)}{1+{J}^{2}\left({\epsilon }\right)},\,{a}_{2}={b}_{1},\end{eqnarray}$where ${a}_{0},$ ${a}_{1},$ and ${a}_{2}$ are unknowns. Substituting the solution (8) into equation (6) and exerting some operations, a system of nonlinear algebraic equations is derived as$\begin{eqnarray*}\begin{array}{l}-4D{a}_{2}{c}_{1}{{\kappa }_{1}}^{2}-4D{a}_{2}{c}_{1}{{\lambda }_{1}}^{2}+2{{a}_{0}}^{3}{c}_{2}{\kappa }_{2}+2{{a}_{0}}^{3}{c}_{3}{\lambda }_{2}\\ \,+\,6{{a}_{0}}^{2}{a}_{2}{c}_{2}{\kappa }_{2}+6{{a}_{0}}^{2}{a}_{2}{c}_{3}{\lambda }_{2}+6{a}_{0}{{a}_{2}}^{2}{c}_{2}{\kappa }_{2}\\ \,+\,6{a}_{0}{{a}_{2}}^{2}{c}_{3}{\lambda }_{2}+2{{a}_{2}}^{3}{c}_{2}{\kappa }_{2}+2{{a}_{2}}^{3}{c}_{3}{\lambda }_{2}-{a}_{0}{c}_{1}{{\kappa }_{2}}^{2}\\ \,-\,{a}_{0}{c}_{1}{{\lambda }_{2}}^{2}-{a}_{2}{c}_{1}{{\kappa }_{2}}^{2}-{a}_{2}{c}_{1}{{\lambda }_{2}}^{2}-{a}_{0}{\mu }_{2}-{a}_{2}{\mu }_{2}=0,\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}-6D{a}_{1}{c}_{1}{{\kappa }_{1}}^{2}-6D{a}_{1}{c}_{1}{{\lambda }_{1}}^{2}+E{a}_{1}{c}_{1}{{\kappa }_{1}}^{2}+E{a}_{1}{c}_{1}{{\lambda }_{1}}^{2}\\ \,+\,6{{a}_{0}}^{2}{a}_{1}{c}_{2}{\kappa }_{2}+6{{a}_{0}}^{2}{a}_{1}{c}_{3}{\lambda }_{2}+12{a}_{0}{a}_{1}{a}_{2}{c}_{2}{\kappa }_{2}\\ \,+\,12{a}_{0}{a}_{1}{a}_{2}{c}_{3}{\lambda }_{2}+6{a}_{1}{{a}_{2}}^{2}{c}_{2}{\kappa }_{2}+6{a}_{1}{{a}_{2}}^{2}{c}_{3}{\lambda }_{2}\\ \,-\,{a}_{1}{c}_{1}{{\kappa }_{2}}^{2}-{a}_{1}{c}_{1}{{\lambda }_{2}}^{2}-{a}_{1}{\mu }_{2}=0,\end{array}\,\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}12D{a}_{2}{c}_{1}{{\kappa }_{1}}^{2}+12D{a}_{2}{c}_{1}{{\lambda }_{1}}^{2}-8E{a}_{2}{c}_{1}{{\kappa }_{1}}^{2}-8E{a}_{2}{c}_{1}{{\lambda }_{1}}^{2}\\ \,+\,6{{a}_{0}}^{3}{c}_{2}{\kappa }_{2}+6{{a}_{0}}^{3}{c}_{3}{\lambda }_{2}+6{{a}_{0}}^{2}{a}_{2}{c}_{2}{\kappa }_{2}+6{{a}_{0}}^{2}{a}_{2}{c}_{3}{\lambda }_{2}\\ \,+\,6{a}_{0}{{a}_{1}}^{2}{c}_{2}{\kappa }_{2}+6{a}_{0}{{a}_{1}}^{2}{c}_{3}{\lambda }_{2}-6{a}_{0}{{a}_{2}}^{2}{c}_{2}{\kappa }_{2}\\ \,-\,6{a}_{0}{{a}_{2}}^{2}{c}_{3}{\lambda }_{2}+6{{a}_{1}}^{2}{a}_{2}{c}_{2}{\kappa }_{2}+6{{a}_{1}}^{2}{a}_{2}{c}_{3}{\lambda }_{2}\\ \,-\,6{{a}_{2}}^{3}{c}_{2}{\kappa }_{2}-6{{a}_{2}}^{3}{c}_{3}{\lambda }_{2}-3{a}_{0}{c}_{1}{{\kappa }_{2}}^{2}-3{a}_{0}{c}_{1}{{\lambda }_{2}}^{2}\\ \,-\,{a}_{2}{c}_{1}{{\kappa }_{2}}^{2}-{a}_{2}{c}_{1}{{\lambda }_{2}}^{2}-3{a}_{0}{\mu }_{2}-{a}_{2}{\mu }_{2}=0,\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}2D{a}_{1}{c}_{1}{{\kappa }_{1}}^{2}+2D{a}_{1}{c}_{1}{{\lambda }_{1}}^{2}-6E{a}_{1}{c}_{1}{{\kappa }_{1}}^{2}-6E{a}_{1}{c}_{1}{{\lambda }_{1}}^{2}\\ \,+\,2F{a}_{1}{c}_{1}{{\kappa }_{1}}^{2}+2F{a}_{1}{c}_{1}{{\lambda }_{1}}^{2}+12{{a}_{0}}^{2}{a}_{1}{c}_{2}{\kappa }_{2}\\ \,+\,12{{a}_{0}}^{2}{a}_{1}{c}_{3}{\lambda }_{2}+2{{a}_{1}}^{3}{c}_{2}{\kappa }_{2}+2{{a}_{1}}^{3}{c}_{3}{\lambda }_{2}-12{a}_{1}{{a}_{2}}^{2}{c}_{2}{\kappa }_{2}\\ \,-\,12{a}_{1}{{a}_{2}}^{2}{c}_{3}{\lambda }_{2}-2{a}_{1}{c}_{1}{{\kappa }_{2}}^{2}-2{a}_{1}{c}_{1}{{\lambda }_{2}}^{2}-2{a}_{1}{\mu }_{2}=0,\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}8E{a}_{2}{c}_{1}{{\kappa }_{1}}^{2}+8E{a}_{2}{c}_{1}{{\lambda }_{1}}^{2}-12F{a}_{2}{c}_{1}{{\kappa }_{1}}^{2}-12F{a}_{2}{c}_{1}{{\lambda }_{1}}^{2}\\ \,+\,6{{a}_{0}}^{3}{c}_{2}{\kappa }_{2}+6{{a}_{0}}^{3}{c}_{3}{\lambda }_{2}-6{{a}_{0}}^{2}{a}_{2}{c}_{2}{\kappa }_{2}-6{{a}_{0}}^{2}{a}_{2}{c}_{3}{\lambda }_{2}\\ \,+\,6{a}_{0}{{a}_{1}}^{2}{c}_{2}{\kappa }_{2}+6{a}_{0}{{a}_{1}}^{2}{c}_{3}{\lambda }_{2}-6{a}_{0}{{a}_{2}}^{2}{c}_{2}{\kappa }_{2}\\ \,-\,6{a}_{0}{{a}_{2}}^{2}{c}_{3}{\lambda }_{2}-6{{a}_{1}}^{2}{a}_{2}{c}_{2}{\kappa }_{2}-6{{a}_{1}}^{2}{a}_{2}{c}_{3}{\lambda }_{2}\\ \,+\,6{{a}_{2}}^{3}{c}_{2}{\kappa }_{2}+6{{a}_{2}}^{3}{c}_{3}{\lambda }_{2}-3{a}_{0}{c}_{1}{{\kappa }_{2}}^{2}-3{a}_{0}{c}_{1}{{\lambda }_{2}}^{2}\\ \,+\,{a}_{2}{c}_{1}{{\kappa }_{2}}^{2}+{a}_{2}{c}_{1}{{\lambda }_{2}}^{2}-3{a}_{0}{\mu }_{2}+{a}_{2}{\mu }_{2}=0,\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}E{a}_{1}{c}_{1}{{\kappa }_{1}}^{2}+E{a}_{1}{c}_{1}{{\lambda }_{1}}^{2}-6F{a}_{1}{c}_{1}{{\kappa }_{1}}^{2}-6F{a}_{1}{c}_{1}{{\lambda }_{1}}^{2}\\ \,+\,6{{a}_{0}}^{2}{a}_{1}{c}_{2}{\kappa }_{2}+6{{a}_{0}}^{2}{a}_{1}{c}_{3}{\lambda }_{2}-12{a}_{0}{a}_{1}{a}_{2}{c}_{2}{\kappa }_{2}\\ \,-\,12{a}_{0}{a}_{1}{a}_{2}{c}_{3}{\lambda }_{2}+6{a}_{1}{{a}_{2}}^{2}{c}_{2}{\kappa }_{2}+6{a}_{1}{{a}_{2}}^{2}{c}_{3}{\lambda }_{2}\\ \,-\,{a}_{1}{c}_{1}{{\kappa }_{2}}^{2}-{a}_{1}{c}_{1}{{\lambda }_{2}}^{2}-{a}_{1}{\mu }_{2}=0,\end{array}\,\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}4F{a}_{2}{c}_{1}{{\kappa }_{1}}^{2}+4F{a}_{2}{c}_{1}{{\lambda }_{1}}^{2}+2{{a}_{0}}^{3}{c}_{2}{\kappa }_{2}+2{{a}_{0}}^{3}{c}_{3}{\lambda }_{2}\\ \,-\,6{{a}_{0}}^{2}{a}_{2}{c}_{2}{\kappa }_{2}-6{{a}_{0}}^{2}{a}_{2}{c}_{3}{\lambda }_{2}+6{a}_{0}{{a}_{2}}^{2}{c}_{2}{\kappa }_{2}\\ \,+\,6{a}_{0}{{a}_{2}}^{2}{c}_{3}{\lambda }_{2}-2{{a}_{2}}^{3}{c}_{2}{\kappa }_{2}-2{{a}_{2}}^{3}{c}_{3}{\lambda }_{2}-{a}_{0}{c}_{1}{{\kappa }_{2}}^{2}\\ \,-\,{a}_{0}{c}_{1}{{\lambda }_{2}}^{2}+{a}_{2}{c}_{1}{{\kappa }_{2}}^{2}+{a}_{2}{c}_{1}{{\lambda }_{2}}^{2}-{a}_{0}{\mu }_{2}+{a}_{2}{\mu }_{2}=0,\end{array}\end{eqnarray*}$whose solution leads to the following cases:

When $D=1,$ $E=-\left({m}^{2}+1\right),$ and $F={m}^{2},$ one derives

$\begin{eqnarray*}\begin{array}{l}\bullet \,{a}_{0}=0,{a}_{1}=\pm 4\sqrt{-\displaystyle \frac{{c}_{1}{{\kappa }_{1}}^{2}+{c}_{1}{{\lambda }_{1}}^{2}}{{c}_{2}{\kappa }_{2}+{c}_{3}{\lambda }_{2}}},{a}_{2}=0,\\ \,\,{\mu }_{2}=-\left(8{{\kappa }_{1}}^{2}+{{\kappa }_{2}}^{2}+8{{\lambda }_{1}}^{2}+{{\lambda }_{2}}^{2}\right){c}_{1},m=1,\end{array}\,\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}\bullet \,{a}_{0}=0,{a}_{1}=0,{a}_{2}=\pm 2\sqrt{-\displaystyle \frac{-{c}_{1}{{\kappa }_{1}}^{2}-{c}_{1}{{\lambda }_{1}}^{2}}{{c}_{2}{\kappa }_{2}+{c}_{3}{\lambda }_{2}}},\\ \,\,{\mu }_{2}=\left(4{{\kappa }_{1}}^{2}-{{\kappa }_{2}}^{2}+4{{\lambda }_{1}}^{2}-{{\lambda }_{2}}^{2}\right){c}_{1},m=1,\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}\bullet \,{a}_{0}=0,{a}_{1}=2\sqrt{-\displaystyle \frac{{c}_{1}{{\kappa }_{1}}^{2}+{c}_{1}{{\lambda }_{1}}^{2}}{{c}_{2}{\kappa }_{2}+{c}_{3}{\lambda }_{2}}},\\ \,\,{a}_{2}=\pm \,\sqrt{-\displaystyle \frac{-{c}_{1}{{\kappa }_{1}}^{2}-{c}_{1}{{\lambda }_{1}}^{2}}{{c}_{2}{\kappa }_{2}+{c}_{3}{\lambda }_{2}}},\\ \,\,{\mu }_{2}=-\left(2{{\kappa }_{1}}^{2}+{{\kappa }_{2}}^{2}+2{{\lambda }_{1}}^{2}+{{\lambda }_{2}}^{2}\right){c}_{1},m=1.\end{array}\end{eqnarray*}$

Now, the exact solutions of the 2D-CNLSE can be constructed as follows$\begin{eqnarray*}\begin{array}{l}{u}_{1,2}\left(x,y,t\right)=\pm 4\sqrt{-\displaystyle \frac{{c}_{1}{{\kappa }_{1}}^{2}+{c}_{1}{{\lambda }_{1}}^{2}}{{c}_{2}{\kappa }_{2}+{c}_{3}{\lambda }_{2}}}\\ \,\times \,\displaystyle \frac{\tanh \left({\kappa }_{1}x+{\lambda }_{1}y-2{c}_{1}\left({\kappa }_{1}{\kappa }_{2}+{\lambda }_{1}{\lambda }_{2}\right)t\right)}{1+{\tanh }^{2}\left({\kappa }_{1}x+{\lambda }_{1}y-2{c}_{1}\left({\kappa }_{1}{\kappa }_{2}+{\lambda }_{1}{\lambda }_{2}\right)t\right)}\\ \,\times \,{{\rm{e}}}^{{\rm{i}}\left({\kappa }_{2}x+{\lambda }_{2}y-\left(8{{\kappa }_{1}}^{2}+{{\kappa }_{2}}^{2}+8{{\lambda }_{1}}^{2}+{{\lambda }_{2}}^{2}\right){c}_{1}t\right)},\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{u}_{3,4}\left(x,y,t\right)=\pm 2\sqrt{-\displaystyle \frac{-{c}_{1}{{\kappa }_{1}}^{2}-{c}_{1}{{\lambda }_{1}}^{2}}{{c}_{2}{\kappa }_{2}+{c}_{3}{\lambda }_{2}}}\\ \,\times \,\displaystyle \frac{1-{\tanh }^{2}\left({\kappa }_{1}x+{\lambda }_{1}y-2{c}_{1}\left({\kappa }_{1}{\kappa }_{2}+{\lambda }_{1}{\lambda }_{2}\right)t\right)}{1+{\tanh }^{2}\left({\kappa }_{1}x+{\lambda }_{1}y-2{c}_{1}\left({\kappa }_{1}{\kappa }_{2}+{\lambda }_{1}{\lambda }_{2}\right)t\right)}\\ \,\times \,{{\rm{e}}}^{{\rm{i}}\left({\kappa }_{2}x+{\lambda }_{2}y+\left(4{{\kappa }_{1}}^{2}-{{\kappa }_{2}}^{2}+4{{\lambda }_{1}}^{2}-{{\lambda }_{2}}^{2}\right){c}_{1}t\right)},\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{u}_{5,6}\left(x,y,t\right)=\left(2\sqrt{-\displaystyle \frac{{c}_{1}{{\kappa }_{1}}^{2}+{c}_{1}{{\lambda }_{1}}^{2}}{{c}_{2}{\kappa }_{2}+{c}_{3}{\lambda }_{2}}}\right.\\ \,\times \,\displaystyle \frac{\tanh \left({\kappa }_{1}x+{\lambda }_{1}y-2{c}_{1}\left({\kappa }_{1}{\kappa }_{2}+{\lambda }_{1}{\lambda }_{2}\right)t\right)}{1+{\tanh }^{2}\left({\kappa }_{1}x+{\lambda }_{1}y-2{c}_{1}\left({\kappa }_{1}{\kappa }_{2}+{\lambda }_{1}{\lambda }_{2}\right)t\right)}\\ \,\pm \,\sqrt{-\displaystyle \frac{-{c}_{1}{{\kappa }_{1}}^{2}-{c}_{1}{{\lambda }_{1}}^{2}}{{c}_{2}{\kappa }_{2}+{c}_{3}{\lambda }_{2}}}\\ \left.\,\times \,\displaystyle \frac{1-{\tanh }^{2}\left({\kappa }_{1}x+{\lambda }_{1}y-2{c}_{1}\left({\kappa }_{1}{\kappa }_{2}+{\lambda }_{1}{\lambda }_{2}\right)t\right)}{1+{\tanh }^{2}\left({\kappa }_{1}x+{\lambda }_{1}y-2{c}_{1}\left({\kappa }_{1}{\kappa }_{2}+{\lambda }_{1}{\lambda }_{2}\right)t\right)}\right)\\ \,\times \,{{\rm{e}}}^{{\rm{i}}\left({\kappa }_{2}x+{\lambda }_{2}y-\left(2{{\kappa }_{1}}^{2}+{{\kappa }_{2}}^{2}+2{{\lambda }_{1}}^{2}+{{\lambda }_{2}}^{2}\right){c}_{1}t\right)}.\end{array}\end{eqnarray*}$

When $D=1-{m}^{2},$ $E=2{m}^{2}-1,$ and $F=-{m}^{2},$ one obtains

$\begin{eqnarray*}\begin{array}{l}{a}_{0}=\pm \sqrt{-\displaystyle \frac{-{c}_{1}{{\kappa }_{1}}^{2}-{c}_{1}{{\lambda }_{1}}^{2}}{4{c}_{2}{\kappa }_{2}+4{c}_{3}{\lambda }_{2}}},{a}_{1}=0,\\ {a}_{2}=\pm \displaystyle \frac{1}{2}\displaystyle \frac{{c}_{1}\left({{\kappa }_{1}}^{2}+{{\lambda }_{1}}^{2}\right)}{{c}_{2}{\kappa }_{2}+{c}_{3}{\lambda }_{2}}\displaystyle \frac{1}{\sqrt{-\tfrac{-{c}_{1}{{\kappa }_{1}}^{2}-{c}_{1}{{\lambda }_{1}}^{2}}{4{c}_{2}{\kappa }_{2}+4{c}_{3}{\lambda }_{2}}}},\\ {\mu }_{2}=\displaystyle \frac{1}{2}\left(5{{\kappa }_{1}}^{2}-2{{\kappa }_{2}}^{2}+5{{\lambda }_{1}}^{2}-2{{\lambda }_{2}}^{2}\right){c}_{1},\,m=\displaystyle \frac{1}{2}.\end{array}\end{eqnarray*}$

Now, the exact solutions of the 2D-CNLSE can be established as follows$\begin{eqnarray*}\begin{array}{l}{u}_{7,8}\left(x,y,t\right)=\pm \sqrt{-\displaystyle \frac{-{c}_{1}{{\kappa }_{1}}^{2}-{c}_{1}{{\lambda }_{1}}^{2}}{4{c}_{2}{\kappa }_{2}+4{c}_{3}{\lambda }_{2}}}\pm \displaystyle \frac{1}{2}\displaystyle \frac{{c}_{1}\left({{\kappa }_{1}}^{2}+{{\lambda }_{1}}^{2}\right)}{{c}_{2}{\kappa }_{2}+{c}_{3}{\lambda }_{2}}\\ \,\times \,\displaystyle \frac{1}{\sqrt{-\tfrac{-{c}_{1}{{\kappa }_{1}}^{2}-{c}_{1}{{\lambda }_{1}}^{2}}{4{c}_{2}{\kappa }_{2}+4{c}_{3}{\lambda }_{2}}}}\\ \,\times \,\displaystyle \frac{1-{{\rm{cn}}}^{2}\left({\kappa }_{1}x+{\lambda }_{1}y-2{c}_{1}\left({\kappa }_{1}{\kappa }_{2}+{\lambda }_{1}{\lambda }_{2}\right)t,\tfrac{1}{2}\right)}{1+{{\rm{cn}}}^{2}\left({\kappa }_{1}x+{\lambda }_{1}y-2{c}_{1}\left({\kappa }_{1}{\kappa }_{2}+{\lambda }_{1}{\lambda }_{2}\right)t,\tfrac{1}{2}\right)}\\ \,\times \,{{\rm{e}}}^{{\rm{i}}\left({\kappa }_{2}x+{\lambda }_{2}y+\displaystyle \frac{1}{2}\left(5{{\kappa }_{1}}^{2}-2{{\kappa }_{2}}^{2}+5{{\lambda }_{1}}^{2}-2{{\lambda }_{2}}^{2}\right){c}_{1}t\right)}.\end{array}\end{eqnarray*}$

When $D={m}^{2},$ $E=-\left({m}^{2}+1\right),$ and $F=1,$ one derives

$\begin{eqnarray*}\begin{array}{l}\bullet \,{a}_{0}=0,\,{a}_{1}=\pm 4\sqrt{-\displaystyle \frac{{c}_{1}{{\kappa }_{1}}^{2}+{c}_{1}{{\lambda }_{1}}^{2}}{{c}_{2}{\kappa }_{2}+{c}_{3}{\lambda }_{2}}},\,{a}_{2}=0,\\ \,\,{\mu }_{2}=-\left(8{{\kappa }_{1}}^{2}+{{\kappa }_{2}}^{2}+8{{\lambda }_{1}}^{2}+{{\lambda }_{2}}^{2}\right){c}_{1},\,m=1,\end{array}\,\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}\bullet \,{a}_{0}=0,\,{a}_{1}=0,\,{a}_{2}=\pm 2\sqrt{-\displaystyle \frac{-{c}_{1}{{\kappa }_{1}}^{2}-{c}_{1}{{\lambda }_{1}}^{2}}{{c}_{2}{\kappa }_{2}+{c}_{3}{\lambda }_{2}}},\\ \,\,{\mu }_{2}=\left(4{{\kappa }_{1}}^{2}-{{\kappa }_{2}}^{2}+4{{\lambda }_{1}}^{2}-{{\lambda }_{2}}^{2}\right){c}_{1},\,m=1,\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}\bullet \,{a}_{0}=0,{a}_{1}=2\sqrt{-\displaystyle \frac{{c}_{1}{{\kappa }_{1}}^{2}+{c}_{1}{{\lambda }_{1}}^{2}}{{c}_{2}{\kappa }_{2}+{c}_{3}{\lambda }_{2}}},\\ \,\,{a}_{2}=\pm \sqrt{-\displaystyle \frac{-{c}_{1}{{\kappa }_{1}}^{2}-{c}_{1}{{\lambda }_{1}}^{2}}{{c}_{2}{\kappa }_{2}+{c}_{3}{\lambda }_{2}}},\\ \,\,{\mu }_{2}=-\left(2{{\kappa }_{1}}^{2}+{{\kappa }_{2}}^{2}+2{{\lambda }_{1}}^{2}+{{\lambda }_{2}}^{2}\right){c}_{1},m=1.\end{array}\end{eqnarray*}$

Now, the exact solutions of the 2D-CNLSE can be constructed as follows$\begin{eqnarray*}\begin{array}{l}{u}_{9,10}\left(x,y,t\right)=\pm 4\sqrt{-\displaystyle \frac{{c}_{1}{{\kappa }_{1}}^{2}+{c}_{1}{{\lambda }_{1}}^{2}}{{c}_{2}{\kappa }_{2}+{c}_{3}{\lambda }_{2}}}\\ \,\times \,\displaystyle \frac{\coth \left({\kappa }_{1}x+{\lambda }_{1}y-2{c}_{1}\left({\kappa }_{1}{\kappa }_{2}+{\lambda }_{1}{\lambda }_{2}\right)t\right)}{1+{\coth }^{2}\left({\kappa }_{1}x+{\lambda }_{1}y-2{c}_{1}\left({\kappa }_{1}{\kappa }_{2}+{\lambda }_{1}{\lambda }_{2}\right)t\right)}\\ \,\times \,{{\rm{e}}}^{{\rm{i}}\left({\kappa }_{2}x+{\lambda }_{2}y-\left(8{{\kappa }_{1}}^{2}+{{\kappa }_{2}}^{2}+8{{\lambda }_{1}}^{2}+{{\lambda }_{2}}^{2}\right){c}_{1}t\right)},\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{u}_{11,12}\left(x,y,t\right)=\pm 2\sqrt{-\displaystyle \frac{-{c}_{1}{{\kappa }_{1}}^{2}-{c}_{1}{{\lambda }_{1}}^{2}}{{c}_{2}{\kappa }_{2}+{c}_{3}{\lambda }_{2}}}\\ \,\times \,\displaystyle \frac{1-{\coth }^{2}\left({\kappa }_{1}x+{\lambda }_{1}y-2{c}_{1}\left({\kappa }_{1}{\kappa }_{2}+{\lambda }_{1}{\lambda }_{2}\right)t\right)}{1+{\coth }^{2}\left({\kappa }_{1}x+{\lambda }_{1}y-2{c}_{1}\left({\kappa }_{1}{\kappa }_{2}+{\lambda }_{1}{\lambda }_{2}\right)t\right)}\\ \,\times \,{{\rm{e}}}^{{\rm{i}}\left({\kappa }_{2}x+{\lambda }_{2}y+\left(4{{\kappa }_{1}}^{2}-{{\kappa }_{2}}^{2}+4{{\lambda }_{1}}^{2}-{{\lambda }_{2}}^{2}\right){c}_{1}t\right)},\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}{u}_{13,14}\left(x,y,t\right)=\left(2\sqrt{-\displaystyle \frac{{c}_{1}{{\kappa }_{1}}^{2}+{c}_{1}{{\lambda }_{1}}^{2}}{{c}_{2}{\kappa }_{2}+{c}_{3}{\lambda }_{2}}}\right.\\ \,\times \,\displaystyle \frac{\coth \left({\kappa }_{1}x+{\lambda }_{1}y-2{c}_{1}\left({\kappa }_{1}{\kappa }_{2}+{\lambda }_{1}{\lambda }_{2}\right)t\right)}{1+{\coth }^{2}\left({\kappa }_{1}x+{\lambda }_{1}y-2{c}_{1}\left({\kappa }_{1}{\kappa }_{2}+{\lambda }_{1}{\lambda }_{2}\right)t\right)}\\ \,\pm \,\sqrt{-\displaystyle \frac{-{c}_{1}{{\kappa }_{1}}^{2}-{c}_{1}{{\lambda }_{1}}^{2}}{{c}_{2}{\kappa }_{2}+{c}_{3}{\lambda }_{2}}}\\ \,\times \,\displaystyle \frac{1-{\coth }^{2}\left({\kappa }_{1}x+{\lambda }_{1}y-2{c}_{1}\left({\kappa }_{1}{\kappa }_{2}+{\lambda }_{1}{\lambda }_{2}\right)t\right)}{1+{\coth }^{2}\left({\kappa }_{1}x+{\lambda }_{1}y-2{c}_{1}\left({\kappa }_{1}{\kappa }_{2}+{\lambda }_{1}{\lambda }_{2}\right)t\right)}\\ \,\times \,{{\rm{e}}}^{{\rm{i}}\left({\kappa }_{2}x+{\lambda }_{2}y-\left(2{{\kappa }_{1}}^{2}+{{\kappa }_{2}}^{2}+2{{\lambda }_{1}}^{2}+{{\lambda }_{2}}^{2}\right){c}_{1}t\right)}.\end{array}\end{eqnarray*}$

When $D=-{m}^{2},$ $E=2{m}^{2}-1,$ and $F=1-{m}^{2},$ one obtains

$\begin{eqnarray*}\begin{array}{l}{a}_{0}=\pm \sqrt{-\displaystyle \frac{-{c}_{1}{{\kappa }_{1}}^{2}-{c}_{1}{{\lambda }_{1}}^{2}}{4{c}_{2}{\kappa }_{2}+4{c}_{3}{\lambda }_{2}}},{a}_{1}=0,\\ {a}_{2}=\mp \displaystyle \frac{1}{2}\displaystyle \frac{{c}_{1}\left({{\kappa }_{1}}^{2}+{{\lambda }_{1}}^{2}\right)}{{c}_{2}{\kappa }_{2}+{c}_{3}{\lambda }_{2}}\displaystyle \frac{1}{\sqrt{-\tfrac{-{c}_{1}{{\kappa }_{1}}^{2}-{c}_{1}{{\lambda }_{1}}^{2}}{4{c}_{2}{\kappa }_{2}+4{c}_{3}{\lambda }_{2}}}},\\ {\mu }_{2}=\displaystyle \frac{1}{2}\left(5{{\kappa }_{1}}^{2}-2{{\kappa }_{2}}^{2}+5{{\lambda }_{1}}^{2}-2{{\lambda }_{2}}^{2}\right){c}_{1},m=\displaystyle \frac{1}{2}.\end{array}\end{eqnarray*}$

Now, the exact solutions of the 2D-CNLSE can be established as follows$\begin{eqnarray*}\begin{array}{l}{u}_{15,16}\left(x,y,t\right)=\pm \sqrt{-\displaystyle \frac{-{c}_{1}{{\kappa }_{1}}^{2}-{c}_{1}{{\lambda }_{1}}^{2}}{4{c}_{2}{\kappa }_{2}+4{c}_{3}{\lambda }_{2}}}\mp \displaystyle \frac{1}{2}\displaystyle \frac{{c}_{1}\left({{\kappa }_{1}}^{2}+{{\lambda }_{1}}^{2}\right)}{{c}_{2}{\kappa }_{2}+{c}_{3}{\lambda }_{2}}\\ \,\times \,\displaystyle \frac{1}{\sqrt{-\tfrac{-{c}_{1}{{\kappa }_{1}}^{2}-{c}_{1}{{\lambda }_{1}}^{2}}{4{c}_{2}{\kappa }_{2}+4{c}_{3}{\lambda }_{2}}}}\\ \,\times \,\displaystyle \frac{1-{{\rm{nc}}}^{2}\left({\kappa }_{1}x+{\lambda }_{1}y-2{c}_{1}\left({\kappa }_{1}{\kappa }_{2}+{\lambda }_{1}{\lambda }_{2}\right)t,\tfrac{1}{2}\right)}{1+{{\rm{nc}}}^{2}\left({\kappa }_{1}x+{\lambda }_{1}y-2{c}_{1}\left({\kappa }_{1}{\kappa }_{2}+{\lambda }_{1}{\lambda }_{2}\right)t,\tfrac{1}{2}\right)}\\ \,\times \,{{\rm{e}}}^{{\rm{i}}\left({\kappa }_{2}x+{\lambda }_{2}y+\displaystyle \frac{1}{2}\left(5{{\kappa }_{1}}^{2}-2{{\kappa }_{2}}^{2}+5{{\lambda }_{1}}^{2}-2{{\lambda }_{2}}^{2}\right){c}_{1}t\right)}.\end{array}\end{eqnarray*}$

Figures 1 and 2 present the 3-dimensional and density plots of $\left|{u}_{1}\left(x,y,t\right)\right|$ and $\left|{u}_{3}\left(x,y,t\right)\right|$ for a series of suitable parameters. More precisely, the parameters ${c}_{1}=-0.1,$ ${c}_{2}=0.1,$ ${c}_{3}=0.1,$ ${\kappa }_{1}=0.3,$ ${\kappa }_{2}=0.3,$ ${\lambda }_{1}=-0.3,$ and ${\lambda }_{2}=0.3$ have been used to portray figure 1 while the parameters ${c}_{1}=0.1,$ ${c}_{2}=0.1,$ ${c}_{3}=0.1,$ ${\kappa }_{1}=0.7,$ ${\kappa }_{2}=0.7,$ ${\lambda }_{1}=-0.7,$ and ${\lambda }_{2}=0.7$ have been utilized to depict figure 2. Clearly, figure 1 shows a dark or topological soliton while figure 2 indicates a bright or nontopological soliton.

According to the knowledge of the authors, the results given in the current paper are new and have not been presented previously.

The results presented in the current research work were examined by Maple, confirming their correctness.

Figure 1.

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Figure 1.The 3-dimensional and density plots of $\left|{u}_{1}\left(x,y,t\right)\right|$ for ${c}_{1}=-0.1,$ ${c}_{2}=0.1,$ ${c}_{3}=0.1,$ ${\kappa }_{1}=0.3,$ ${\kappa }_{2}=0.3,$ ${\lambda }_{1}=-0.3,$ ${\lambda }_{2}=0.3,$ and $t=0.$


Figure 2.

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Figure 2.The 3-dimensional and density plots of $\left|{u}_{3}\left(x,y,t\right)\right|$ for ${c}_{1}=0.1,$ ${c}_{2}=0.1,$ ${c}_{3}=0.1,$ ${\kappa }_{1}=0.7,$ ${\kappa }_{2}=0.7,$ ${\lambda }_{1}=-0.7,$ ${\lambda }_{2}=0.7,$ and $t=0.$


4. Conclusion

The main aim of the current article was to study a nonlinear evolution equation referred to as the 2D-CNLSE in mathematical physics. The study firstly progressed with adopting a complex transform to reduce the 2D-CNLSE to a nonlinear ODE in the real domain with a known soliton velocity. The MJEE method was then adopted to obtain soliton and other solutions of the 2D-CNLSE that were classified as topological and nontopological solitons as well as Jacobi elliptic function solutions. The present article provided useful information regarding the 2D-CNLSE and its exact solutions.

Reference By original order
By published year
By cited within times
By Impact factor

Nishino A Umeno Y Wadati M 1998 Chiral nonlinear Schrödinger equation
Chaos, Solitons Fractals 9 1063 1069

DOI:10.1016/S0960-0779(97)00184-7 [Cited within: 1]

Aglietti U Griguolo L Jackiw R Pi S Y Semirara D 1996 Anyons and chiral solitons on a line
Phys. Rev. Lett. 77 4406 4409

DOI:10.1103/PhysRevLett.77.4406 [Cited within: 1]

Biswas A 2009 Chiral solitons in 1+2 dimensions
Int. J. Theor. Phys. 48 3403 3409

DOI:10.1007/s10773-009-0145-4 [Cited within: 3]

Eslami M 2016 Trial solution technique to chiral nonlinear Schrodinger's equation in (1+2)-dimensions
Nonlinear Dyn. 85 813 816

DOI:10.1007/s11071-016-2724-2 [Cited within: 1]

Raza N Javid A 2019 Optical dark and dark-singular soliton solutions of (1+2)-dimensional chiral nonlinear Schrodinger's equation
Waves Random Complex Media 29 496 508

DOI:10.1080/17455030.2018.1451009 [Cited within: 1]

Raza N Arshed S 2020 Chiral bright and dark soliton solutions of Schrödinger's equation in (1+2)-dimensions
Ain Shams Eng. J.

DOI:10.1016/j.asej.2020.03.018 [Cited within: 2]

Ma H C Zhang Z P Deng A P 2012 A new periodic solution to Jacobi elliptic functions of MKdV equation and BBM equation
Acta Math. Appl. Sin. 28 409 415

DOI:10.1007/s10255-012-0153-7 [Cited within: 2]

Hosseini K Mirzazadeh M Osman M S Al Qurashi M Baleanu D 2020 Solitons and Jacobi elliptic function solutions to the complex Ginzburg-Landau equation
Front. Phys. 8 225

DOI:10.3389/fphy.2020.00225 [Cited within: 1]

Zayed E M E Shohib R M A Biswas A Yıldırım Y Mallawi F Belic M R 2019 Chirped and chirp-free solitons in optical fiber Bragg gratings with dispersive reflectivity having parabolic law nonlinearity by Jacobi's elliptic function
Results Phys. 15 102784

DOI:10.1016/j.rinp.2019.102784

Zayed E M E Alngar M E M 2020 Optical solitons in birefringent fibers with Biswas-Arshed model by generalized Jacobi elliptic function expansion method
Optik 203 163922

DOI:10.1016/j.ijleo.2019.163922

Hosseini K Mirzazadeh M Ilie M Gómez-Aguilar J F 2020 Biswas-Arshed equation with the beta time derivative: optical solitons and other solutions
Optik 217 164801

DOI:10.1016/j.ijleo.2020.164801

El-Sheikh M M A Seadawy A R Ahmed H M Arnous A H Rabie W B 2020 Dispersive and propagation of shallow water waves as a higher order nonlinear Boussinesq-like dynamical wave equations
Physica A 537 122662

DOI:10.1016/j.physa.2019.122662

Hosseini K Mirzazadeh M Vahidi J Asghari R 2020 Optical wave structures to the Fokas-Lenells equation
Optik 207 164450

DOI:10.1016/j.ijleo.2020.164450

Hosseini K Matinfar M Mirzazadeh M 2020 A (3+1)-dimensional resonant nonlinear Schrödinger equation and its Jacobi elliptic and exponential function solutions
Optik 207 164458

DOI:10.1016/j.ijleo.2020.164458 [Cited within: 1]

Hosseini K Mirzazadeh M Ilie M Radmehr S 2020 Dynamics of optical solitons in the perturbed Gerdjikov-Ivanov equation
Optik 206 164350

DOI:10.1016/j.ijleo.2020.164350 [Cited within: 1]

Hosseini K Osman M S Mirzazadeh M Rabiei F 2020 Investigation of different wave structures to the generalized third-order nonlinear Scrödinger equation
Optik 206 164259

DOI:10.1016/j.ijleo.2020.164259

Hosseini K Mirzazadeh M Zhou Q Liu Y Moradi M 2019 Analytic study on chirped optical solitons in nonlinear metamaterials with higher order effects
Laser Phys. 29 095402

DOI:10.1088/1555-6611/ab356f

Hosseini K Mirzazadeh M Rabiei F Baskonus H M Yel G 2020 Dark optical solitons to the Biswas-Arshed equation with high order dispersions and absence of self-phase modulation
Optik 209 164576

DOI:10.1016/j.ijleo.2020.164576

Hosseini K Ma W X Ansari R Mirzazadeh M Pouyanmehr R Samadani F 2020 Evolutionary behavior of rational wave solutions to the (4+1)-dimensional Boiti-Leon-Manna-Pempinelli equation
Phys. Scr. 95 065208

DOI:10.1088/1402-4896/ab7fee

Biswas A Arshed S 2018 Optical solitons in presence of higher order dispersions and absence of self-phase modulation
Optik 174 452 459

DOI:10.1016/j.ijleo.2018.08.037

Biswas A Ullah M Z Zhou Q Moshokoa S P Triki H Belic M 2017 Resonant optical solitons with quadratic-cubic nonlinearity by semi-inverse variational principle
Optik 145 18 21

DOI:10.1016/j.ijleo.2017.07.028

Kumar S Malik S Biswas A Yıldırım Y Alshomrani A S Belic M R 2020 Optical solitons with generalized anti-cubic nonlinearity by Lie symmetry
Optik 206 163638

DOI:10.1016/j.ijleo.2019.163638

Yıldırım Y Biswas A Jawad A J M Ekici M Zhou Q Khan S Alzahrani A K Belic M R 2020 Cubic-quartic optical solitons in birefringent fibers with four forms of nonlinear refractive index by exp-function expansion
Results Phys. 16 102913

DOI:10.1016/j.rinp.2019.102913

Kudryashov N A 2020 Method for finding highly dispersive optical solitons of nonlinear differential equations
Optik 206 163550

DOI:10.1016/j.ijleo.2019.163550

Kudryashov N A 2020 Highly dispersive optical solitons of the generalized nonlinear eighth-order Schrödinger equation
Optik 206 164335

DOI:10.1016/j.ijleo.2020.164335

Kudryashov N A 2020 Solitary wave solutions of the generalized Biswas-Arshed equation
Optik 219 165002

DOI:10.1016/j.ijleo.2020.165002

Houwe A Inc M Doka S Y Akinlar M A Baleanu D 2020 Chirped solitons in negative index materials generated by Kerr nonlinearity
Results Phys. 17 103097

DOI:10.1016/j.rinp.2020.103097

Aliyu A I Inc M Yusuf A Baleanu D 2019 Optical solitons and stability analysis with spatio-temporal dispersion in Kerr and quadric-cubic nonlinear media
Optik 178 923 931

DOI:10.1016/j.ijleo.2018.10.046

Inc M Aliyu A I Yusuf A Baleanu D 2018 Combined optical solitary waves and conservation laws for nonlinear Chen-Lee-Liu equation in optical fibers
Optik 158 297 304

DOI:10.1016/j.ijleo.2017.12.075

Srivastava H M Baleanu D Machado J A T Osman M S Rezazadeh R Arshed S Günerhan H 2020 Traveling wave solutions to nonlinear directional couplers by modified Kudryashov method
Phys. Scr. 95 075217

DOI:10.1088/1402-4896/ab95af

Chen Y X Xu F Q Hu Y L 2019 Excitation control for three-dimensional Peregrine solution and combined breather of a partially nonlocal variable-coefficient nonlinear Schrödinger equation
Nonlinear Dyn. 95 1957 1964

DOI:10.1007/s11071-018-4670-7

Dai C Q Fan Y Zhang N 2019 Re-observation on localized waves constructed by variable separation solutions of (1+1)-dimensional coupled integrable dispersionless equations via the projective Riccati equation method
Appl. Math. Lett. 96 20 26

DOI:10.1016/j.aml.2019.04.009

Wu G Z Dai C Q 2020 Nonautonomous soliton solutions of variable-coefficient fractional nonlinear Schrödinger equation
Appl. Math. Lett. 106 106365

DOI:10.1016/j.aml.2020.106365

Wang B H Wang Y Y 2020 Fractional white noise functional soliton solutions of a wick-type stochastic fractional NLSE
Appl. Math. Lett. 110 106583

DOI:10.1016/j.aml.2020.106583

Yu L J Wu G Z Wang Y Y Chen Y X 2020 Traveling wave solutions constructed by Mittag-Leffler function of a (2+1)-dimensional space-time fractional NLS equation
Results Phys. 17 103156

DOI:10.1016/j.rinp.2020.103156

Dai C Q Fan Y Wang Y Y 2019 Three-dimensional optical solitons formed by the balance between different-order nonlinearities and high-order dispersion/diffraction in parity-time symmetric potentials
Nonlinear Dyn. 98 489 499

DOI:10.1007/s11071-019-05206-z [Cited within: 1]

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