Various Solitons and Other Wave Solutions to the (2+1)-Dimensional Heisenberg Ferromagnetic Spin Chain Dynamical Model

Abstract

This paper outlines a study into the exact solutions of the (2+1)-dimensional Heisenberg ferromagnetic spin chain equation that is used to illustrate the ferromagnetic materials of magnetic ordering by applying two recent techniques, namely, the Sardar-subequation method and extended rational sine–cosine and sinh–cosh methods. Abundant exact solutions such as the bright soliton, dark soliton, combined bright–dark soliton, singular soliton and other periodic wave solutions expressed by the generalized trigonometric, generalized hyperbolic, trigonometric and hyperbolic functions are obtained. The numerical results are illustrated in the form of 3D plots, 2D contours and 2D curves by choosing proper parametric values to interpret the physical behavior of the model. The obtained results in this work are expected to provide a rich platform for constructing the soliton solutions of PDEs in physics.

1. Introduction

In recent decades, scholars have proposed many important nonlinear evolution equations to simulate different nonlinear phenomena involved in optics [1,2,3,4,5], plasma physics [6,7], thermodynamics [8], vibration [9,10], quantum mechanics [11,12] and so on. The question of how to obtain the exact solutions of nonlinear partial differential equations has been a hot topic for mathematicians and physicists. So far, many effective and powerful methods have been put forward such as the extended trial equation method [13,14], sine–Gordon expansion method [15,16], generalized (G’/G) expansion method [17,18], extended rational sine–cosine and sinh–cosh method [19,20], Wang’s Bäcklund transformation-based method [21], the extended FAN subequation method [22] and so on [23,24,25,26,27,28,29]. In this paper, we aim to study the (2+1)-dimensional Heisenberg ferromagnetic spin chain equation (HFSCE) controlled by a new integrable (2+1)-dimensional nonlinear Schrödinger-type equation which reads as [30,31]:

$$i{\Xi }_t+{\epsilon }_1{\Xi }_{xx}+{\epsilon }_2{\Xi }_{yy}+{\epsilon }_3{\Xi }_{xy}-{\epsilon }_4{\left|\Xi \right|}^2\Xi =0,$$

(1)

where $\Xi =\Xi \left(x,y,t\right)$ is the complex function of x, y, t, ${\epsilon }_1={\lambda }^4\left(v+v_2\right)$, ${\epsilon }_2={\lambda }^4\left(v_1+v_2\right)$, ${\epsilon }_3=3\lambda v_2$, ${\epsilon }_4=3{\lambda }^{4}A$. The descriptions of these parameters are shown in

Table 1. The descriptions of the coefficients.

Parameter	Description
$\lambda$	The lattice parameter
$v$	the coefficient of bilinear exchange interactions along the $x$-direction
$v_1$	the coefficient of bilinear exchange interactions along the $y$-direction
$v_2$	the neighboring interaction along the diagonal
$A$	the uniaxial crystal field anisotropy paramete

. Equation (1) is usually used to describe the wave propagation in the Heisenberg ferromagnetic spin chain. In [30], two methods are used to seek for diverse solitons, namely the generalized Riccati mapping method and improved auxiliary equation. In [31], the Jacobi elliptic expansion method is adopted and bright and dark soliton solutions are obtained. In [32], the theory of dynamical systems is introduced for the construction of the bounded traveling wave solutions. In [33], the modified Khater method is used to deal with Equation (1). The generalized tanh method is employed to solve Equation (1) in [34], and dark, dark–bright or combined optical and singular soliton solutions are attained. In [35], the authors use two approaches, namely the modified F-Expansion and projective Riccati equation methods, to investigate Equation (1). In [36], the extended FAN subequation method is presented to study Equation (1) and different soliton solutions like the bright, dark, combined bright–dark and other wave solutions are constructed. In our work [37], the variational method and subequation method are used to deal with Equation (1) and different soliton solutions are constructed. As the expansion of our previous achievements, in this study we will seek to obtain the exact solutions of Equation (1) by other new methods, namely the Sardar-subequation method (SSBM) and extended rational sine–cosine and sinh–cosh methods (ERSSM). The structure of this paper is as follows. We provide a detailed introduction of the algorithms of the two methods in Section 2. We apply the two methods to find the exact solutions in Section 3. In Section 4, the behaviors of some solutions and their physical interpretations are given. Finally, we reach a conclusion in Section 5.

2. The Methods

The purpose of this section is to provide a brief introduction to the SSBM and ERSSM.

Consider a general PDE as:

$$f\left(\Xi ,{\Xi }_x,{\Xi }_y,{\Xi }_t,\dots \dots \right)=0.$$

(2)

Making use of the transformation as:

$$\Xi \left(x,t\right)=\Phi \left(\Re \right), \Re =mx+ny-kt$$

(3)

where m, n are the wave numbers, k is the group velocity of the wave.

It can turn Equation (2) into an ODE which reads as:

$$f\left(\Phi ,{\Phi }_{\Re },{\Phi }_{\Re \Re },{\Phi }_{\Re \Re \Re },\dots \dots \right)=0.$$

(4)

2.1. The SSBM

Based on the SSBM [38,39,40], we can suppose Equation (4) has the solution as:

$$\Phi \left(\Re \right)={\displaystyle \sum _{i=0}^s{a}_i{\xi }^i}\left(\Re \right).$$

(5)

where ${\alpha }_i\left(i=0,1,2\dots ,s\right)$ are the constants to be determined later. There is:

$${\left({\xi }^{\prime }\right)}^2=p+q{\xi }^2+{\xi }^4,$$

(6)

where ${\xi }^{\prime }=\frac{d\xi }{d\Re }$, $p$, $q$ are constants. Equation (6) has the following solutions for different conditions:

Case I: When $q>0$ and $p=0$, we have:

$${\xi }_1^{\pm }\left(\Re \right)=\pm \sqrt{\varphi \phi q}{\mathrm{sech}}_{\varphi \phi }\left(\sqrt{q}\Re \right),$$

$${\xi }_2^{\pm }\left(\Re \right)=\pm \sqrt{\varphi \phi q}{\mathrm{csch}}_{\varphi \phi }\left(\sqrt{q}\Re \right),$$

where ${\mathrm{sech}}_{\varphi \phi }\left(\Re \right)=\frac{2}{\varphi e^{\Re }+\phi e^{-\Re }}$, ${\mathrm{csch}}_{\varphi \phi }\left(\Re \right)=\frac{2}{\varphi e^{\Re }-\phi e^{-\Re }}$.

Case II: When $q<0$ and $p=0$, we have:

$${\xi }_3^{\pm }\left(\Re \right)=\pm \sqrt{-\varphi \phi q}{\sec }_{\varphi \phi }\left(\sqrt{-q}\Re \right),$$

$${\xi }_4^{\pm }\left(\Re \right)=\pm \sqrt{-\varphi \phi q}{\csc }_{\varphi \phi }\left(\sqrt{-q}\Re \right),$$

where ${\sec }_{\varphi \phi }\left(\Re \right)=\frac{2}{\varphi e^{i\Re }+\phi e^{-i\Re }}$, ${\mathrm{csch}}_{\varphi \phi }\left(\Re \right)=\frac{2i}{\varphi e^{i\Re }-\phi e^{-i\Re }}$.

Case III: When $q>0$ and $p=\frac{q^2}{4}$, we have:

$${\xi }_5^{\pm }\left(\Re \right)=\pm \sqrt{\frac{q}{2}}{\tan }_{\varphi \phi }\left(\sqrt{\frac{q}{2}}\Re \right),$$

$${\xi }_6^{\pm }\left(\Re \right)=\pm \sqrt{\frac{q}{2}}{\cot }_{\varphi \phi }\left(\sqrt{\frac{q}{2}}\Re \right),$$

$$\begin{gathered} {\xi }_7^{\pm }\left(\Re \right)=\pm \sqrt{\frac{q}{2}}\left[{\tan }_{\varphi \phi }\left(\sqrt{2q}\Re \right)\pm \sqrt{\varphi \phi }{\sec }_{\varphi \phi }\left(\sqrt{2q}\Re \right)\right], \\ {\xi }_8^{\pm }\left(\Re \right)=\pm \sqrt{\frac{q}{2}}\left[{\cot }_{\varphi \phi }\left(\sqrt{2q}\Re \right)\pm \sqrt{\varphi \phi }{\csc }_{\varphi \phi }\left(\sqrt{2q}\Re \right)\right], \end{gathered}$$

$${\xi }_9^{\pm }\left(\Re \right)=\pm \sqrt{\frac{q}{8}}\left[{\tan }_{\varphi \phi }\left(\sqrt{\frac{q}{8}}\Re \right)+{\cot }_{\varphi \phi }\left(\sqrt{\frac{q}{8}}\Re \right)\right],$$

where ${\tan }_{\varphi \phi }\left(\Re \right)=-i\frac{\varphi e^{i\Re }-\phi e^{-i\Re }}{\varphi e^{i\Re }+\phi e^{-i\Re }}$, ${\cot }_{\varphi \phi }\left(\Re \right)=i\frac{\varphi e^{i\Re }+\phi e^{-i\Re }}{\varphi e^{i\Re }-\phi e^{-i\Re }}$.

Case IV: When $q<0$ and $p=\frac{q^2}{4}$, we have:

$${\xi }_{10}^{\pm }\left(\Re \right)=\pm \sqrt{-\frac{q}{2}}{\tanh }_{\varphi \phi }\left(\sqrt{-\frac{q}{2}}\Re \right),$$

$${\xi }_{11}^{\pm }\left(\Re \right)=\pm \sqrt{-\frac{q}{2}}{\coth }_{\varphi \phi }\left(\sqrt{-\frac{q}{2}}\Re \right),$$

$$\begin{gathered} {\xi }_{12}^{\pm }\left(\Re \right)=\pm \sqrt{-\frac{q}{2}}\left[{\tanh }_{\varphi \phi }\left(\sqrt{-2q}\Re \right)\pm \sqrt{\varphi \phi }{\mathrm{sech}}_{\varphi \phi }\left(\sqrt{-2q}\Re \right)\right], \\ {\xi }_{13}^{\pm }\left(\Re \right)=\pm \sqrt{-\frac{q}{2}}\left[{\coth }_{\varphi \phi }\left(\sqrt{-2q}\Re \right)\pm \sqrt{\varphi \phi }{\mathrm{csch}}_{\varphi \phi }\left(\sqrt{-2q}\Re \right)\right], \end{gathered}$$

$${\xi }_{14}^{\pm }\left(\Re \right)=\pm \sqrt{-\frac{q}{8}}\left[{\tanh }_{\varphi \phi }\left(\sqrt{-\frac{q}{8}}\Re \right)+{\coth }_{\varphi \phi }\left(\sqrt{-\frac{q}{8}}\Re \right)\right],$$

where ${\tanh }_{\varphi \phi }\left(\Re \right)=\frac{\varphi e^{\Re }-\phi e^{-\Re }}{\varphi e^{\Re }+\phi e^{-\Re }}$, ${\coth }_{\varphi \phi }\left(\Re \right)=\frac{\varphi e^{\Re }+\phi e^{-\Re }}{\varphi e^{\Re }-\phi e^{-\Re }}$.

2.2. The ERSSM

(1) The extended rational sine–cosine method

Supposing the solution of Equation (4) is [19]:

$$\Phi \left(\Re \right)=\frac{{\mu }_0\sin \left(\sigma \Re \right)}{{\mu }_1\cos \left(\sigma \Re \right)+{\mu }_2},$$

(7)

Or

$$\Phi \left(\Re \right)=\frac{{\mu }_0\cos \left(\sigma \Re \right)}{{\mu }_1\sin \left(\sigma \Re \right)+{\mu }_2},$$

(8)

where ${\mu }_0$, ${\mu }_1$ and ${\mu }_2$ are unknown constants to be determined.

Taking Equation (7) or Equation (8) into Equation (4), making the corresponding adjustments and setting the coefficients of ${\cos }^i\left(\sigma \Re \right)$ or ${\sin }^i\left(\sigma \Re \right)$ as zero yields a system of algebraic equations. Solving it, the unknown coefficients can be obtained.

(2) The extended rational sinh–cosh method

The solution of Equation (4) is considered as [41]:

$$\Phi \left(\Re \right)=\frac{{\mu }_0\sinh \left(\sigma \Re \right)}{{\mu }_1\cosh \left(\sigma \Re \right)+{\mu }_2},$$

(9)

Or

$$\Phi \left(\Re \right)=\frac{{\mu }_0\cosh \left(\sigma \Re \right)}{{\mu }_1\sinh \left(\sigma \Re \right)+{\mu }_2},$$

(10)

By the same means, we obtain a system of algebraic equations by equating the coefficients of $\cosh \left(\sigma \Re \right)$ or $\sinh \left(\sigma \Re \right)$ to zero in this case. Solving the system, we can determine the unknown coefficients.

3. Applications

The following complex transformation is introduced:

$$\Xi =\Phi \left(\Re \right)e^{i\chi }, \Re =x+y-\eta t, \chi =mx+ny-kt.$$

(11)

Taking Equation (11) into Equation (1) produces the following the imaginary and real parts, respectively as:

$$\eta =2{\epsilon }_{1}m+2{\epsilon }_{2}n+{\epsilon }_3\left(m+n\right),$$

(12)

$$\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right){\Phi }^{″}+\left[k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)\right]\Phi -{\epsilon }_4{\Phi }^3=0.$$

(13)

where ${\Phi }^{″}=\frac{d^2\Phi }{d{\Re }^2}$. We re-write Equation (13) as:

$${\Phi }^{″}+{\Im }_1\Phi -{\Im }_2{\Phi }^3=0,$$

(14)

where

$${\Im }_1=\frac{k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}, {\Im }_2=\frac{{\epsilon }_4}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}.$$

(15)

3.1. Application of the Sardar-Subequation Method

The solution of Equation (14) is assumed as:

$$\Phi \left(\Re \right)={\displaystyle \sum _{i=0}^s{a}_i{\xi }^i}\left(\Re \right),$$

(16)

A balance between ${\Phi }^{″}$ and ${\Phi }^3$ gives $s=1$, so Equation (16) reduces to:

$$\Phi \left(\Re \right)=a_0+a_1\xi \left(\Re \right),$$

(17)

Inserting Equation (17) with Equation (6) into Equation (14) and adjusting the obtained results yields:

$$\begin{gathered} {\xi }^0: p{a}_0-q{a}_0^3=0, \\ {\xi }^1: q{a}_1+{\Im }_1{a}_1-3{\Im }_2{a}_0^2{a}_1=0, \\ {\xi }^2:-3{\Im }_2{a}_0{a}_1^2=0, \\ {\xi }^3:2a_1^{}-{\Im }_2{a}_1^3=0, \end{gathered}$$

Solving them, we obtain:

$$a_0=0, a_1=\pm \sqrt{\frac{2}{{\Im }_2}}, q=-{\Im }_1,$$

(18)

Taking Equation (15) into it has:

$$a_0=0, a_1=\pm \sqrt{\frac{2\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)}{{\epsilon }_4}}, q=\frac{{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3},$$

So, the solutions of Equation (1) are:

Case I: If $\frac{{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}>0$ and $p=0$, we have:

$${\Xi }_1^{\pm }\left(x,y,t\right)=\pm \sqrt{\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)\varphi \phi }{{\epsilon }_4}}{\mathrm{sech}}_{\varphi \phi }\left(\sqrt{\frac{{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)e^{i\left(mx+ny-kt\right)},$$

(19)

which is the bright soliton solution for $\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)\varphi \phi {\epsilon }_4>0$ and $\left[{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right]\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)>0$.

$${\Xi }_2^{\pm }\left(x,y,t\right)=\pm \sqrt{\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)\varphi \phi }{{\epsilon }_4}}{\mathrm{csch}}_{\varphi \phi }\left(\sqrt{\frac{{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)e^{i\left(mx+ny-kt\right)},$$

(20)

which represents the singular soliton solution for $\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)\varphi \phi {\epsilon }_4>0$ and $\left[{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right]\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)>0$.

Case II: If $\frac{{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}<0$ and $p=0$, the singular periodic wave solutions is as:

$${\Xi }_3^{\pm }\left(x,y,t\right)=\pm \sqrt{-\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)\varphi \phi }{{\epsilon }_4}}{\sec }_{\varphi \phi }\left(\sqrt{-\frac{{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)e^{i\left(mx+ny-kt\right)},$$

(21)

and

$${\Xi }_4^{\pm }\left(x,y,t\right)=\pm \sqrt{-\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)\varphi \phi }{{\epsilon }_4}}{\csc }_{\varphi \phi }\left(\sqrt{-\frac{{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)e^{i\left(mx+ny-kt\right)},$$

(22)

which are both under the conditions for $\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)\varphi \phi {\epsilon }_4<0$ and $\left[{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right]\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)<0$.

Case III: If $\frac{{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}>0$ and $p=\frac{{\left[{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right]}^2}{4{\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)}^2}$, Equation (1) formats the singular periodic wave solutions as:

$${\Xi }_5^{\pm }\left(x,y,t\right)=\pm \sqrt{\frac{\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_4}}{\tan }_{\varphi \phi }\left(\sqrt{\frac{{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k}{2\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)}}\left(x+y-\eta t\right)\right)e^{i\left(mx+ny-kt\right)},$$

(23)

$${\Xi }_6^{\pm }\left(x,y,t\right)=\pm \sqrt{\frac{\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_4}}{\cot }_{\varphi \phi }\left(\sqrt{\frac{{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k}{2\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)}}\left(x+y-\eta t\right)\right)e^{i\left(mx+ny-kt\right)},$$

(24)

$$\begin{array}{l}{\Xi }_7^{\pm }\left(x,y,t\right)=\pm \sqrt{\frac{\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_4}}\left[\begin{array}{l}{\tan }_{\varphi \phi }\left(\sqrt{\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)\\ \pm \sqrt{\varphi \phi }{\sec }_{\varphi \phi }\left(\sqrt{\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)\end{array}\right]\\ \begin{array}{ccccccccccc}& & & & & & & & & & \times \end{array}{e}^{i\left(mx+ny-kt\right)}\end{array}$$

(25)

$$\begin{array}{l}{\Xi }_8^{\pm }\left(x,y,t\right)=\pm \sqrt{\frac{\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_4}}\left[\begin{array}{l}{\cot }_{\varphi \phi }\left(\sqrt{\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)\\ \pm \sqrt{\varphi \phi }{\csc }_{\varphi \phi }\left(\sqrt{\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)\end{array}\right]\\ \begin{array}{ccccccccccc}& & & & & & & & & & \times \end{array}{e}^{i\left(mx+ny-kt\right)}\end{array}$$

(26)

$${\Xi }_9^{\pm }\left(x,y,t\right)=\pm \sqrt{\frac{\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{4{\epsilon }_4}}\left[\begin{array}{l}{\tan }_{\varphi \phi }\left(\sqrt{\frac{{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k}{8\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)}}\left(x+y-\eta t\right)\right)\\ +{\cot }_{\varphi \phi }\left(\sqrt{\frac{{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k}{8\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)}}\left(x+y-\eta t\right)\right)\end{array}\right]e^{i\left(mx+ny-kt\right)},$$

(27)

which are all under the conditions for $\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right){\epsilon }_4>0$ and $\left[{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right]\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)>0$.

Case IV: When $\frac{{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}<0$ and $p=\frac{{\left[{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right]}^2}{4{\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)}^2}$, Equation (1) has the following different soliton solutions under the conditions for $\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right){\epsilon }_4<0$ and $\left[{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right]\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)<0$:

$${\Xi }_{10}^{\pm }\left(x,y,t\right)=\pm \sqrt{-\frac{\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_4}}{\tanh }_{\varphi \phi }\left(\sqrt{-\frac{{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k}{2\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)}}\left(x+y-\eta t\right)\right)e^{i\left(mx+ny-kt\right)},$$

(28)

$${\Xi }_{11}^{\pm }\left(x,y,t\right)=\pm \sqrt{-\frac{\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_4}}{\coth }_{\varphi \phi }\left(\sqrt{-\frac{{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k}{2\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)}}\left(x+y-\eta t\right)\right)e^{i\left(mx+ny-kt\right)},$$

(29)

$$\begin{array}{l}{\Xi }_{12}^{\pm }\left(x,y,t\right)=\pm \sqrt{-\frac{\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_4}}\left[\begin{array}{l}{\tanh }_{\varphi \phi }\left(\sqrt{-\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)\\ \pm \sqrt{\varphi \phi }{\mathrm{sech}}_{\varphi \phi }\left(\sqrt{-\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)\end{array}\right]\\ \begin{array}{cccccccccccc}& & & & & & & & & & & \times \end{array}{e}^{i\left(mx+ny-kt\right)}\end{array}$$

(30)

$$\begin{array}{l}{\Xi }_{13}^{\pm }\left(x,y,t\right)=\pm \sqrt{-\frac{\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_4}}\left[\begin{array}{l}{\coth }_{\varphi \phi }\left(\sqrt{-\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)\\ \pm \sqrt{\varphi \phi }{\mathrm{csch}}_{\varphi \phi }\left(\sqrt{-\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)\end{array}\right]\\ \begin{array}{cccccccccccc}& & & & & & & & & & & \times \end{array}{e}^{i\left(mx+ny-kt\right)}\end{array}$$

(31)

$${\Xi }_{14}^{\pm }\left(x,y,t\right)=\pm \sqrt{-\frac{\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{4{\epsilon }_4}}\left[\begin{array}{l}{\tanh }_{\varphi \phi }\left(\sqrt{-\frac{{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k}{8\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)}}\left(x+y-\eta t\right)\right)\\ +{\coth }_{\varphi \phi }\left(\sqrt{-\frac{{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k}{8\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)}}\left(x+y-\eta t\right)\right)\end{array}\right]e^{i\left(mx+ny-kt\right)}.$$

(32)

where Equation (28) is the dark soliton solution, Equations (29)–(32) represent the singular soliton solution, and Equation (30) means the dark–bright soliton solution.

3.2. Application of the Extended Rational Sine–Cosine and Sinh–Cosh Methods

Supposing the solution of Equation (14) as

$$\Phi \left(\Re \right)=\frac{{\mu }_0\sin \left(\sigma \Re \right)}{{\mu }_1\cos \left(\sigma \Re \right)+{\mu }_2},$$

(33)

Plugging it into Equation (14) and taking the proper treatments gives:

$\cos {\left(\sigma \Re \right)}^0$: $-{\Im }_2{\mu }_0^2+2{\sigma }^2{\mu }_1^2+{\Im }_1{\mu }_2^2-{\sigma }^2{\mu }_2^2=0$,

$\cos {\left(\sigma \Re \right)}^1$: $2{\Im }_1{\mu }_1^{}{\mu }_2^{}+{\sigma }^2{\mu }_1^{}{\mu }_2^{}=0$,

$\cos {\left(\sigma \Re \right)}^2$: ${\Im }_2{\mu }_0^2+{\Im }_1{\mu }_1^2=0$,

Solving them, we obtain:

Case I: ${\mu }_0=\pm \sqrt{\frac{{\Im }_1}{{\Im }_2}}{\mu }_2$, ${\mu }_1=\pm {\mu }_2$, $\sigma =\pm \sqrt{-2{\Im }_1}$,

In the view of Equation (15), we have the singular periodic wave solutions of Equation (1) as:

$${\Xi }_{15}^{\pm }\left(x,y,t\right)=\frac{\pm \sqrt{\frac{k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)}{{\epsilon }_4}}\sin \left(-\sqrt{\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)}{-\cos \left(-\sqrt{\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)+1}{e}^{i\left(mx+ny-kt\right)},$$

(34)

$${\Xi }_{16}^{\pm }\left(x,y,t\right)=\frac{\pm \sqrt{\frac{k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)}{{\epsilon }_4}}\sin \left(-\sqrt{\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)}{\cos \left(-\sqrt{\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)+1}{e}^{i\left(mx+ny-kt\right)},$$

(35)

$${\Xi }_{17}^{\pm }\left(x,y,t\right)=\frac{\pm \sqrt{\frac{k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)}{{\epsilon }_4}}\sin \left(\sqrt{\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)}{-\cos \left(\sqrt{\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)+1}{e}^{i\left(mx+ny-kt\right)},$$

(36)

$${\Xi }_{18}^{\pm }\left(x,y,t\right)=\frac{\pm \sqrt{\frac{k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)}{{\epsilon }_4}}\sin \left(\sqrt{\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)}{\cos \left(\sqrt{\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)+1}{e}^{i\left(mx+ny-kt\right)},$$

(37)

Case II: ${\mu }_0=\pm \sqrt{\frac{{\Im }_1}{{\Im }_2}}{\mu }_1$, ${\mu }_2=0$, $\sigma =\pm \sqrt{-\frac{{\Im }_1}{2}}$,

In the view of Equation (15), Equation (1) has the singular periodic wave solutions as:

$${\Xi }_{19}^{\pm }\left(x,y,t\right)=\pm \sqrt{\frac{k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)}{{\epsilon }_4}}\tan \left(\sqrt{\frac{{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)+k}{2\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)}}\left(x+y-\eta t\right)\right)e^{i\left(mx+ny-kt\right)},$$

(38)

Assume Equation (14) has the solution as:

$$\Phi \left(\Re \right)=\frac{{\mu }_0\cos \left(\sigma \Re \right)}{{\mu }_1\sin \left(\sigma \Re \right)+{\mu }_2},$$

(39)

Similarly, we have:

$\sin {\left(\sigma \Re \right)}^0$: $-{\Im }_2{\mu }_0^2+2{\sigma }^2{\mu }_1^2+{\Im }_1{\mu }_2^2-{\sigma }^2{\mu }_2^2=0$,

$\sin {\left(\sigma \Re \right)}^1$: $2{\Im }_1{\mu }_1^{}{\mu }_2^{}+{\sigma }^2{\mu }_1^{}{\mu }_2^{}=0$,

$\sin {\left(\sigma \Re \right)}^2$: ${\Im }_2{\mu }_0^2+{\Im }_1{\mu }_1^2=0$,

Solving above equations gives:

Case I: ${\mu }_0=\pm \sqrt{\frac{{\Im }_1}{{\Im }_2}}{\mu }_2$, ${\mu }_1=\pm {\mu }_2$, $\sigma =\pm \sqrt{-2{\Im }_1}$,

In the view of Equation (15), we have the singular periodic wave solutions of Equation (1) as:

$${\Xi }_{20}^{\pm }\left(x,y,t\right)=\frac{\pm \sqrt{\frac{k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)}{{\epsilon }_4}}\cos \left(-\sqrt{\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)}{-\sin \left(-\sqrt{\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)+1}{e}^{i\left(mx+ny-kt\right)},$$

(40)

$${\Xi }_{21}^{\pm }\left(x,y,t\right)=\frac{\pm \sqrt{\frac{k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)}{{\epsilon }_4}}\cos \left(-\sqrt{\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)}{\sin \left(-\sqrt{\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)+1}{e}^{i\left(mx+ny-kt\right)},$$

(41)

$${\Xi }_{22}^{\pm }\left(x,y,t\right)=\frac{\pm \sqrt{\frac{k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)}{{\epsilon }_4}}\cos \left(\sqrt{\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)}{-\sin \left(\sqrt{\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)+1}{e}^{i\left(mx+ny-kt\right)},$$

(42)

$${\Xi }_{23}^{\pm }\left(x,y,t\right)=\frac{\pm \sqrt{\frac{k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)}{{\epsilon }_4}}\cos \left(\sqrt{\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)}{\sin \left(\sqrt{\frac{2\left({\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)+1}{e}^{i\left(mx+ny-kt\right)},$$

(43)

Case II: ${\mu }_0=\pm \sqrt{\frac{{\Im }_1}{{\Im }_2}}{\mu }_1$, ${\mu }_2=0$, $\sigma =\pm \sqrt{-\frac{{\Im }_1}{2}}$,

In the view of Equation (15), we have the singular periodic wave solutions as:

$${\Xi }_{24}^{\pm }\left(x,y,t\right)=\pm \sqrt{\frac{k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)}{{\epsilon }_4}}\cot \left(\sqrt{\frac{{\epsilon }_1{m}^2+n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)-k}{2\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)}}\left(x+y-\eta t\right)\right)e^{i\left(mx+ny-kt\right)},$$

(44)

We can also assume the solution of Equation (14) as:

$$\Phi \left(\Re \right)=\frac{{\mu }_0\sinh \left(\sigma \Re \right)}{{\mu }_1\cosh \left(\sigma \Re \right)+{\mu }_2},$$

(45)

Putting it into Equation (14) yields:

$\cosh {\left(\sigma \Re \right)}^0$: ${\Im }_2{\mu }_0^2-2{\sigma }^2{\mu }_1^2+{\Im }_1{\mu }_2^2+{\sigma }^2{\mu }_2^2=0$,

$\cosh {\left(\sigma \Re \right)}^1$: $2{\Im }_1{\mu }_1^{}{\mu }_2^{}-{\sigma }^2{\mu }_1^{}{\mu }_2^{}=0$,

$\cosh {\left(\sigma \Re \right)}^2$: $-{\Im }_2{\mu }_0^2+{\Im }_1{\mu }_1^2=0$,

Solving them produces:

Case I: ${\mu }_0=\pm \sqrt{\frac{{\Im }_1}{{\Im }_2}}{\mu }_2$, ${\mu }_1=\pm {\mu }_2$, $\sigma =\pm \sqrt{2{\Im }_1}$,

In the view of Equation (15), we obtain the following soliton solutions of Equation (1) as:

$${\Xi }_{25}^{\pm }\left(x,y,t\right)=\pm \sqrt{\frac{k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)}{{\epsilon }_4}}\tanh \left(\sqrt{\frac{\left(k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)\right)}{2\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)}}\left(x+y-\eta t\right)\right)e^{i\left(mx+ny-kt\right)},$$

(46)

which means that the dark soliton solution for $\left[k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)\right]{\epsilon }_4>0$ and $\left[k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)\right]\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)>0$.

$${\Xi }_{26}^{\pm }\left(x,y,t\right)=\pm \sqrt{\frac{k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)}{{\epsilon }_4}}\coth \left(\sqrt{\frac{\left(k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)\right)}{2\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)}}\left(x+y-\eta t\right)\right)e^{i\left(mx+ny-kt\right)},$$

(47)

which is the singular soliton solution for $\left[k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)\right]{\epsilon }_4>0$ and $\left[k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)\right]\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)>0$.

Case II: ${\mu }_0=\pm \sqrt{\frac{{\Im }_1}{{\Im }_2}}{\mu }_1$, ${\mu }_2=0$, $\sigma =\pm \sqrt{\frac{{\Im }_1}{2}}$,

In the light of Equation (15), we obtain the dark soliton solution for Equation (1) as:

$${\Xi }_{27}^{\pm }\left(x,y,t\right)=\pm \sqrt{\frac{k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)}{{\epsilon }_4}}\tanh \left(\sqrt{\frac{k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)}{2\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)}}\left(x+y-\eta t\right)\right)e^{i\left(mx+ny-kt\right)},$$

(48)

which is under the condition for $\left[k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)\right]{\epsilon }_4>0$ and $\left[k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)\right]\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)>0$.

We can also assume the solution of Equation (14) as:

$$\Phi \left(\Re \right)=\frac{{\mu }_0\cosh \left(\sigma \Re \right)}{{\mu }_1\sinh \left(\sigma \Re \right)+{\mu }_2},$$

(49)

In the same manner, we have:

$\sinh {\left(\sigma \Re \right)}^0$: ${\Im }_2{\mu }_0^2-2{\sigma }^2{\mu }_1^2+{\Im }_1{\mu }_2^2+{\sigma }^2{\mu }_2^2=0$,

$\sinh {\left(\sigma \Re \right)}^1$: $2{\Im }_1{\mu }_1^{}{\mu }_2^{}-{\sigma }^2{\mu }_1^{}{\mu }_2^{}=0$,

$\sinh {\left(\sigma \Re \right)}^2$: $-{\Im }_2{\mu }_0^2+{\Im }_1{\mu }_1^2=0$,

Solving them produces:

Case I: ${\mu }_0=\pm \sqrt{\frac{{\Im }_1}{{\Im }_2}}{\mu }_2$, ${\mu }_1=\pm {\mu }_2$, $\sigma =\pm \sqrt{2{\Im }_1}$,

In the view of Equation (15), we have the singular soliton solutions for Equation (1) as:

$${\Xi }_{28}^{\pm }\left(x,y,t\right)=\frac{\pm \sqrt{\frac{k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)}{{\epsilon }_4}}\cosh \left(-\sqrt{\frac{2\left(k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)}{-\sinh \left(-\sqrt{\frac{2\left(k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)+1}{e}^{i\left(mx+ny-kt\right)},$$

(50)

$${\Xi }_{29}^{\pm }\left(x,y,t\right)=\frac{\pm \sqrt{\frac{k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)}{{\epsilon }_4}}\cosh \left(-\sqrt{\frac{2\left(k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)}{\sinh \left(-\sqrt{\frac{2\left(k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)+1}{e}^{i\left(mx+ny-kt\right)},$$

(51)

$${\Xi }_{30}^{\pm }\left(x,y,t\right)=\frac{\pm \sqrt{\frac{k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)}{{\epsilon }_4}}\cosh \left(\sqrt{\frac{2\left(k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)}{-\sinh \left(\sqrt{\frac{2\left(k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)+1}{e}^{i\left(mx+ny-kt\right)},$$

(52)

$${\Xi }_{31}^{\pm }\left(x,y,t\right)=\frac{\pm \sqrt{\frac{k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)}{{\epsilon }_4}}\cosh \left(\sqrt{\frac{2\left(k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)}{\sinh \left(\sqrt{\frac{2\left(k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)\right)}{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3}}\left(x+y-\eta t\right)\right)+1}{e}^{i\left(mx+ny-kt\right)},$$

(53)

which are all under the conditions for $\left[k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)\right]{\epsilon }_4>0$ and $\left[k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)\right]\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)>0$.

Case II: ${\mu }_0=\pm \sqrt{\frac{{\Im }_1}{{\Im }_2}}{\mu }_1$, ${\mu }_2=0$, $\sigma =\pm \sqrt{\frac{{\Im }_1}{2}}$,

In the light of Equation (15), we have the singular soliton solutions for Equation (1) as:

$${\Xi }_{32}^{\pm }\left(x,y,t\right)=\pm \sqrt{\frac{k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)}{{\epsilon }_4}}\coth \left(\sqrt{\frac{k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)}{2\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)}}\left(x+y-\eta t\right)\right)e^{i\left(mx+ny-kt\right)}.$$

(54)

which are both under the conditions for $\left[k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)\right]{\epsilon }_4>0$ and $\left[k-{\epsilon }_1{m}^2-n\left({\epsilon }_{2}n+{\epsilon }_{3}m\right)\right]\left({\epsilon }_1+{\epsilon }_2+{\epsilon }_3\right)>0$.

By comparing the solutions of Equations (46) and (47) obtained by the ERSSM with the solutions of Equations (28) and (29) for $\varphi =1$ and $\phi =1$, which were attained by the SSBE, respectively, good agreement is reached, which reveals that the proposed methods are correct and effective. In addition, a comparison between the obtained results and our previous works [39] reveals that Equations (28) and (46) (for $\varphi =1$ and $\phi =1$) are the same as the solution Equation (46) obtained in [37] for $a=1$, $b=1$. Equations (29) and (47) (for $\varphi =1$ and $\phi =1$) are the same as the solution Equation (47) reported in [37] for $a=1$, $b=1$.

4. Discussion and the Physical Interpretations

The purpose of this section is to present the behaviors of the soliton solutions graphically with the help of the Mathematica software.

We illustrate the behaviors of $|{\Xi }_1^{+}\left(x,y,t\right){|}^2$, ${\left|{\Xi }_2^{+}\left(x,y,t\right)\right|}^2$, ${\left|{\Xi }_{10}^{+}\left(x,y,t\right)\right|}^2$, ${\left|{\Xi }_{12}^{+}\left(x,y,t\right)\right|}^2$ and ${\left|{\Xi }_{27}^{+}\left(x,y,t\right)\right|}^2$ via the 3D plot, 2D contour and 2D curve to understand the physical behavior of the model by choosing the proper parameters in Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5. We can find that the wave form of $|{\Xi }_1^{+}\left(x,y,t\right){|}^2$ is the bright soliton, the wave form of ${\left|{\Xi }_2^{+}\left(x,y,t\right)\right|}^2$ is the singular soliton, the profiles of the ${\left|{\Xi }_{10}^{+}\left(x,y,t\right)\right|}^2$ and ${\left|{\Xi }_{27}^{+}\left(x,y,t\right)\right|}^2$ are both the dark solitons and the profile of ${\left|{\Xi }_{12}^{+}\left(x,y,t\right)\right|}^2$ is the combined dark–bright soliton.

5. Conclusions

Two effective approaches, namely the Sardar-subequation method and extended rational sine–cosine and sinh–cosh methods, have been used in this work to search for the exact solutions of the (2+1)-dimensional HFSCE. Various soliton solutions, such as the bright soliton, dark soliton, combined dark–bright soliton, singular soliton and other periodic wave solutions are constructed and expressed in the form of generalized trigonometric, generalized hyperbolic, trigonometric and hyperbolic functions. Numerical simulation of the different soliton solutions is presented via the 3D plots, 2D contours and 2D curves to interpret the physical behavior of the model by assigning appropriate parameters. The comparison between the solutions obtained by the SSBE and those obtained by the ERSSM shows that the solutions Equations (46) and (47) are the same as those for Equations (28) and (29), respectively for $\varphi =1$ and $\phi =1$, which strongly proves the correctness and effectiveness of the proposed methods.

Recently, the fractal and fractional calculus [42,43,44,45,46] has received public attention in many fields. The methods proposed in this work can be employed to study the soliton and travelling wave solutions of the fractal and fractional PDEs arising in the field of physics.