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. Jacobi elliptic function solutions of equation ( |
| No. | D | E | F | J(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 | $-{m}^{2}$ | $2{m}^{2}-1$ | $1-{m}^{2}$ | ${\rm{nc}}\left(\xi \right)$ |
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 follows8 ) into equation (6 ) and exerting some operations, a system of nonlinear algebraic equations is derived as
$\begin{eqnarray}U\left({\epsilon }\right)={a}_{0}+{a}_{1}\displaystyle \frac{J\left({\epsilon }\right)}{1+{J}^{2}\left({\epsilon }\right)}+{a}_{2}\displaystyle \frac{1-{J}^{2}\left({\epsilon }\right)}{1+{J}^{2}\left({\epsilon }\right)},\,{a}_{2}={b}_{1},\end{eqnarray}$
where ${a}_{0},$ ${a}_{1},$ and ${a}_{2}$ are unknowns. Substituting the solution ( $\begin{eqnarray*}\begin{array}{l}-4D{a}_{2}{c}_{1}{{\kappa }_{1}}^{2}-4D{a}_{2}{c}_{1}{{\lambda }_{1}}^{2}+2{{a}_{0}}^{3}{c}_{2}{\kappa }_{2}+2{{a}_{0}}^{3}{c}_{3}{\lambda }_{2}\\ \,+\,6{{a}_{0}}^{2}{a}_{2}{c}_{2}{\kappa }_{2}+6{{a}_{0}}^{2}{a}_{2}{c}_{3}{\lambda }_{2}+6{a}_{0}{{a}_{2}}^{2}{c}_{2}{\kappa }_{2}\\ \,+\,6{a}_{0}{{a}_{2}}^{2}{c}_{3}{\lambda }_{2}+2{{a}_{2}}^{3}{c}_{2}{\kappa }_{2}+2{{a}_{2}}^{3}{c}_{3}{\lambda }_{2}-{a}_{0}{c}_{1}{{\kappa }_{2}}^{2}\\ \,-\,{a}_{0}{c}_{1}{{\lambda }_{2}}^{2}-{a}_{2}{c}_{1}{{\kappa }_{2}}^{2}-{a}_{2}{c}_{1}{{\lambda }_{2}}^{2}-{a}_{0}{\mu }_{2}-{a}_{2}{\mu }_{2}=0,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}-6D{a}_{1}{c}_{1}{{\kappa }_{1}}^{2}-6D{a}_{1}{c}_{1}{{\lambda }_{1}}^{2}+E{a}_{1}{c}_{1}{{\kappa }_{1}}^{2}+E{a}_{1}{c}_{1}{{\lambda }_{1}}^{2}\\ \,+\,6{{a}_{0}}^{2}{a}_{1}{c}_{2}{\kappa }_{2}+6{{a}_{0}}^{2}{a}_{1}{c}_{3}{\lambda }_{2}+12{a}_{0}{a}_{1}{a}_{2}{c}_{2}{\kappa }_{2}\\ \,+\,12{a}_{0}{a}_{1}{a}_{2}{c}_{3}{\lambda }_{2}+6{a}_{1}{{a}_{2}}^{2}{c}_{2}{\kappa }_{2}+6{a}_{1}{{a}_{2}}^{2}{c}_{3}{\lambda }_{2}\\ \,-\,{a}_{1}{c}_{1}{{\kappa }_{2}}^{2}-{a}_{1}{c}_{1}{{\lambda }_{2}}^{2}-{a}_{1}{\mu }_{2}=0,\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}12D{a}_{2}{c}_{1}{{\kappa }_{1}}^{2}+12D{a}_{2}{c}_{1}{{\lambda }_{1}}^{2}-8E{a}_{2}{c}_{1}{{\kappa }_{1}}^{2}-8E{a}_{2}{c}_{1}{{\lambda }_{1}}^{2}\\ \,+\,6{{a}_{0}}^{3}{c}_{2}{\kappa }_{2}+6{{a}_{0}}^{3}{c}_{3}{\lambda }_{2}+6{{a}_{0}}^{2}{a}_{2}{c}_{2}{\kappa }_{2}+6{{a}_{0}}^{2}{a}_{2}{c}_{3}{\lambda }_{2}\\ \,+\,6{a}_{0}{{a}_{1}}^{2}{c}_{2}{\kappa }_{2}+6{a}_{0}{{a}_{1}}^{2}{c}_{3}{\lambda }_{2}-6{a}_{0}{{a}_{2}}^{2}{c}_{2}{\kappa }_{2}\\ \,-\,6{a}_{0}{{a}_{2}}^{2}{c}_{3}{\lambda }_{2}+6{{a}_{1}}^{2}{a}_{2}{c}_{2}{\kappa }_{2}+6{{a}_{1}}^{2}{a}_{2}{c}_{3}{\lambda }_{2}\\ \,-\,6{{a}_{2}}^{3}{c}_{2}{\kappa }_{2}-6{{a}_{2}}^{3}{c}_{3}{\lambda }_{2}-3{a}_{0}{c}_{1}{{\kappa }_{2}}^{2}-3{a}_{0}{c}_{1}{{\lambda }_{2}}^{2}\\ \,-\,{a}_{2}{c}_{1}{{\kappa }_{2}}^{2}-{a}_{2}{c}_{1}{{\lambda }_{2}}^{2}-3{a}_{0}{\mu }_{2}-{a}_{2}{\mu }_{2}=0,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}2D{a}_{1}{c}_{1}{{\kappa }_{1}}^{2}+2D{a}_{1}{c}_{1}{{\lambda }_{1}}^{2}-6E{a}_{1}{c}_{1}{{\kappa }_{1}}^{2}-6E{a}_{1}{c}_{1}{{\lambda }_{1}}^{2}\\ \,+\,2F{a}_{1}{c}_{1}{{\kappa }_{1}}^{2}+2F{a}_{1}{c}_{1}{{\lambda }_{1}}^{2}+12{{a}_{0}}^{2}{a}_{1}{c}_{2}{\kappa }_{2}\\ \,+\,12{{a}_{0}}^{2}{a}_{1}{c}_{3}{\lambda }_{2}+2{{a}_{1}}^{3}{c}_{2}{\kappa }_{2}+2{{a}_{1}}^{3}{c}_{3}{\lambda }_{2}-12{a}_{1}{{a}_{2}}^{2}{c}_{2}{\kappa }_{2}\\ \,-\,12{a}_{1}{{a}_{2}}^{2}{c}_{3}{\lambda }_{2}-2{a}_{1}{c}_{1}{{\kappa }_{2}}^{2}-2{a}_{1}{c}_{1}{{\lambda }_{2}}^{2}-2{a}_{1}{\mu }_{2}=0,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}8E{a}_{2}{c}_{1}{{\kappa }_{1}}^{2}+8E{a}_{2}{c}_{1}{{\lambda }_{1}}^{2}-12F{a}_{2}{c}_{1}{{\kappa }_{1}}^{2}-12F{a}_{2}{c}_{1}{{\lambda }_{1}}^{2}\\ \,+\,6{{a}_{0}}^{3}{c}_{2}{\kappa }_{2}+6{{a}_{0}}^{3}{c}_{3}{\lambda }_{2}-6{{a}_{0}}^{2}{a}_{2}{c}_{2}{\kappa }_{2}-6{{a}_{0}}^{2}{a}_{2}{c}_{3}{\lambda }_{2}\\ \,+\,6{a}_{0}{{a}_{1}}^{2}{c}_{2}{\kappa }_{2}+6{a}_{0}{{a}_{1}}^{2}{c}_{3}{\lambda }_{2}-6{a}_{0}{{a}_{2}}^{2}{c}_{2}{\kappa }_{2}\\ \,-\,6{a}_{0}{{a}_{2}}^{2}{c}_{3}{\lambda }_{2}-6{{a}_{1}}^{2}{a}_{2}{c}_{2}{\kappa }_{2}-6{{a}_{1}}^{2}{a}_{2}{c}_{3}{\lambda }_{2}\\ \,+\,6{{a}_{2}}^{3}{c}_{2}{\kappa }_{2}+6{{a}_{2}}^{3}{c}_{3}{\lambda }_{2}-3{a}_{0}{c}_{1}{{\kappa }_{2}}^{2}-3{a}_{0}{c}_{1}{{\lambda }_{2}}^{2}\\ \,+\,{a}_{2}{c}_{1}{{\kappa }_{2}}^{2}+{a}_{2}{c}_{1}{{\lambda }_{2}}^{2}-3{a}_{0}{\mu }_{2}+{a}_{2}{\mu }_{2}=0,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}E{a}_{1}{c}_{1}{{\kappa }_{1}}^{2}+E{a}_{1}{c}_{1}{{\lambda }_{1}}^{2}-6F{a}_{1}{c}_{1}{{\kappa }_{1}}^{2}-6F{a}_{1}{c}_{1}{{\lambda }_{1}}^{2}\\ \,+\,6{{a}_{0}}^{2}{a}_{1}{c}_{2}{\kappa }_{2}+6{{a}_{0}}^{2}{a}_{1}{c}_{3}{\lambda }_{2}-12{a}_{0}{a}_{1}{a}_{2}{c}_{2}{\kappa }_{2}\\ \,-\,12{a}_{0}{a}_{1}{a}_{2}{c}_{3}{\lambda }_{2}+6{a}_{1}{{a}_{2}}^{2}{c}_{2}{\kappa }_{2}+6{a}_{1}{{a}_{2}}^{2}{c}_{3}{\lambda }_{2}\\ \,-\,{a}_{1}{c}_{1}{{\kappa }_{2}}^{2}-{a}_{1}{c}_{1}{{\lambda }_{2}}^{2}-{a}_{1}{\mu }_{2}=0,\end{array}\,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}4F{a}_{2}{c}_{1}{{\kappa }_{1}}^{2}+4F{a}_{2}{c}_{1}{{\lambda }_{1}}^{2}+2{{a}_{0}}^{3}{c}_{2}{\kappa }_{2}+2{{a}_{0}}^{3}{c}_{3}{\lambda }_{2}\\ \,-\,6{{a}_{0}}^{2}{a}_{2}{c}_{2}{\kappa }_{2}-6{{a}_{0}}^{2}{a}_{2}{c}_{3}{\lambda }_{2}+6{a}_{0}{{a}_{2}}^{2}{c}_{2}{\kappa }_{2}\\ \,+\,6{a}_{0}{{a}_{2}}^{2}{c}_{3}{\lambda }_{2}-2{{a}_{2}}^{3}{c}_{2}{\kappa }_{2}-2{{a}_{2}}^{3}{c}_{3}{\lambda }_{2}-{a}_{0}{c}_{1}{{\kappa }_{2}}^{2}\\ \,-\,{a}_{0}{c}_{1}{{\lambda }_{2}}^{2}+{a}_{2}{c}_{1}{{\kappa }_{2}}^{2}+{a}_{2}{c}_{1}{{\lambda }_{2}}^{2}-{a}_{0}{\mu }_{2}+{a}_{2}{\mu }_{2}=0,\end{array}\end{eqnarray*}$
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. 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. 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.