Stochastic Solitons in Birefringent Fibers for Biswas–Arshed Equation with Multiplicative White Noise via Itô Calculus by Modified Extended Mapping Method

Abstract

Stochastic partial differential equations have wide applications in various fields of science and engineering. This paper addresses the optical stochastic solitons and other exact stochastic solutions through birefringent fibers for the Biswas–Arshed equation with multiplicative white noise using the modified extended mapping method. This model contains many kinds of soliton solutions, which are always symmetric or anti-symmetric in space. Stochastic bright soliton solutions, stochastic dark soliton solutions, stochastic combo bright–dark soliton solutions, stochastic combo singular-bright soliton solutions, stochastic singular soliton solutions, stochastic periodic solutions, stochastic rational solutions, stochastic Weierstrass elliptic doubly periodic solutions, and stochastic Jacobi elliptic function solutions are extracted. The constraints on the parameters are considered to guarantee the existence of these stochastic solutions. Furthermore, some of the selected solutions are described graphically to demonstrate the physical nature of the obtained solutions.

1. Introduction

The noise arising in physics, telecommunication networks, hydrology, medicine, and so on can be modeled by stochastic partial differential equations. One of the important research directions of stochastic partial differential equations is the stochastic wave, which has been studied by many authors (see [1,2,3]). Optical solitons are used to transmit very large amounts of data at the speed of light without error and loss over very long distances (see [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. However, some natural conditions cause stochastic distortions in communication. Noise cannot be ignored because its effects will lead to distortions and significant consequences. For this reason, it is important to establish differential equations with stochastic terms so that the established models can be more precise. Subsequently, the solitary wave solutions of nonlinear stochastic partial differential equations (NSPDEs) are always symmetric or anti-symmetric in space. Some of the literature discusses the stochastic soliton solutions of NSPDEs, for example, Khan et al. [32] studied the stochastic perturbation of optical solitons with anti-cubic nonlinearity with bandpass filters and multi-photon absorption. He and Wang [33] obtained dark multi-soliton and soliton molecule solutions of the stochastic nonlinear Schrödinger equation in the white noise space. Arshed et al. [34] studied the chiral solitons of the (2+1)-dimensional stochastic chiral nonlinear Schrödinger equation. Secer [35] investigated the stochastic optical solitons with multiplicative white noise via Itô calculus. Yin et al. [36] discussed the stochastic soliton solutions for the (2+1)-dimensional stochastic Broer–Kaup equations in a fluid or plasma. Zayed et al. [37] extracted optical solitons in birefringent fibers with the Biswas–Arshed equation with multiplicative noise via Itô calculus using the extended simplest equation algorithm and the Jacobi elliptic expansion approach. Saleh et al. [38] discussed Lie symmetry analysis of stochastic gene evolution in a double-chain deoxyribonucleic acid system.

In the present study, we consider the Biswas–Arshed equation with multiplicative white noise in the Itô sense in the following form [37]:

$$i{\varphi }_t+{\wp }_{11}{\varphi }_{xx}+{\wp }_{12}{\varphi }_{xt}+i\left({\wp }_{13}{\varphi }_{xxx}+{\wp }_{14}{\varphi }_{xxt}\right)+{\mathcal{L}}_1\left(\varphi -i{\wp }_{12}{\varphi }_x+{\wp }_{14}{\varphi }_{xx}\right)\frac{d{\omega }_1}{dt}=$$

$$i\left[{\alpha }_1{\left({\left|\mathsf{\Psi }\right|}^2\right)}_x\varphi +{\beta }_1{\left|\mathsf{\Psi }\right|}^2{\varphi }_x+{\theta }_1{\left|\varphi \right|}^2{\varphi }_x+{\lambda }_1{\left(\varphi {\left|\varphi \right|}^2\right)}_x+{\mu }_1{\left({\left|\varphi \right|}^2\right)}_x\varphi +{{\rm Y}}_1{\left(\varphi {\left|\mathsf{\Psi }\right|}^2\right)}_x\right],$$

(1)

$$i{\mathsf{\Psi }}_t+{\wp }_{21}{\mathsf{\Psi }}_{xx}+{\wp }_{22}{\mathsf{\Psi }}_{xt}+i\left({\wp }_{23}{\mathsf{\Psi }}_{xxx}+{\wp }_{24}{\mathsf{\Psi }}_{xxt}\right)+{\mathcal{L}}_2\left(\mathsf{\Psi }-i{\wp }_{22}{\mathsf{\Psi }}_x+{\wp }_{24}{\mathsf{\Psi }}_{xx}\right)\frac{d{\omega }_2}{dt}=$$

$$i\left[{\alpha }_2{\left({\left|\varphi \right|}^2\right)}_x\mathsf{\Psi }+{\beta }_2{\left|\varphi \right|}^2{\mathsf{\Psi }}_x+{\theta }_2{\left|\mathsf{\Psi }\right|}^2{\mathsf{\Psi }}_x+{\lambda }_2{\left(\mathsf{\Psi }{\left|\mathsf{\Psi }\right|}^2\right)}_x+{\mu }_2{\left({\left|\mathsf{\Psi }\right|}^2\right)}_x\mathsf{\Psi }+{{\rm Y}}_2{\left(\mathsf{\Psi }{\left|\varphi \right|}^2\right)}_x\right],$$

(2)

where the complex-valued functions are denoted by $\varphi$ and $\mathsf{\Psi }$, ${\wp }_{11}$ and ${\wp }_{21}$ are the coefficients of chromatic dispersion, ${\wp }_{12}$ and ${\wp }_{22}$ are the coefficients of spatio-temporal dispersion, ${\wp }_{13}$ and ${\wp }_{23}$ are the coefficients of third-order dispersion, and ${\wp }_{14}$ and ${\wp }_{24}$ are the coefficients of third-order spatio-temporal dispersion. ${\mathcal{L}}_1$ and ${\mathcal{L}}_2$ are the coefficients of the noise strength and ${\omega }_1$ and ${\omega }_2$ are the standard Wiener processes, such that $\frac{d{\omega }_1}{dt}$ and $\frac{d{\omega }_2}{dt}$ are the white noise. ${\lambda }_1$ and ${\lambda }_2$ are the coefficients of self-steepening. The coefficients of the nonlinear dispersion terms are represented by ${\alpha }_i,{\beta }_i,{\theta }_i,{\mu }_i,{{\rm Y}}_i(i=1,2)$.

In this paper, the modified extended mapping method is presented for the proposed model to obtain stochastic optical solitons through birefringent fibers and other stochastic exact solutions. The presented method gives a variety of solutions and the stochastic bright soliton solutions, stochastic dark soliton solutions, stochastic combo bright–dark soliton solutions, stochastic combo singular-bright soliton solutions, stochastic singular soliton solutions, stochastic periodic solutions, stochastic rational solutions, stochastic Weierstrass elliptic doubly periodic solutions, and stochastic Jacobi elliptic function solutions are extracted. Our motivation is to improve the work of [37] by giving many new solutions that can help in the field of telecommunications for coding or the transmission of data. When we compare our results and those obtained in [37] using the extended simplest equation algorithm and the Jacobi elliptic expansion approach, we observe that there are many new solutions in the present work. Moreover, some of the obtained stochastic solutions are represented graphically.

2. Preliminaries

In this section, the modified extended mapping method is presented briefly [39,40]. Assume the following nonlinear partial differential equation:

$$F\left(\varphi ,{\varphi }_t,{\varphi }_{x_1},{\varphi }_{x_1{x}_2},{\varphi }_{t{x}_1},...\right)=0.$$

(3)

Suppose that Equation (3) has a traveling wave solution of the form:

$$\varphi (t,x_1,x_2,x_3,\dots ,x_n)=\mathcal{Y}\left(\zeta \right),\zeta =\sum _{i=0}^n{x}_i-\Im t.$$

(4)

Then, Equation (3) is reduced to the following nonlinear ordinary differential equation:

$$\mathcal{H}(\mathcal{Y},{\mathcal{Y}}^{\prime },{\mathcal{Y}}^{″},\dots )=0.$$

(5)

We suppose that Equation (5) has a solution of the following form:

$$\mathcal{Y}\left(\zeta \right)=\sum _{j=0}^N{\mathit{d}}_j\chi {\left(\zeta \right)}^j+\sum _{j=-1}^{-N}{\mathit{f}}_{-j}\chi {\left(\zeta \right)}^j+\sum _{j=-1}^{-N}{\mathit{h}}_{-j}{\chi }^{\prime }\left(\zeta \right)\chi {\left(\zeta \right)}^j+\sum _{j=2}^N{\mathit{r}}_j{\chi }^{\prime }\left(\zeta \right)\chi {\left(\zeta \right)}^{j-2},$$

(6)

where ${\mathit{d}}_j$,${\mathit{f}}_{-j}$,${\mathit{h}}_{-j}$, and ${\mathit{r}}_j$ are constants, and $\chi \left(\zeta \right)$ achieves the following equation:

$${\chi }^{\prime }\left(\zeta \right)=\sqrt{{\rho }_0+{\rho }_1\chi \left(\zeta \right)+{\rho }_2\chi {\left(\zeta \right)}^2+{\rho }_3\chi {\left(\zeta \right)}^3+{\rho }_4\chi {\left(\zeta \right)}^4+{\rho }_6\chi {\left(\zeta \right)}^6},$$

(7)

where ${\rho }_l,(l=0,1,2,3,4,6)$ are constants.

Through Equation (5), the positive integer N is found by balancing the highest-order nonlinear terms and the highest-order derivatives.

Solution (6) is substituted into Equation (5) with the auxiliary Equation (7). Then, the coefficients of the same powers are summed and set equal to zero, then, we obtain a system of algebraic equations that can be solved by Mathematica or Maple to obtain the unknown constants ${\mathit{d}}_j$,${\mathit{f}}_{-j}$,${\mathit{h}}_{-j}$, ${\mathit{r}}_j,{\rho }_i$, and ℑ. Thus, we obtain the exact solutions of Equation (3).

3. Stochastic Solitons and Other Solutions

In order to obtain the stochastic solutions of (1) and (2), we use the following wave transformations:

$$\varphi (x,t)={\mathcal{Y}}_1\left(\zeta \right)e^{i\left(Z_1(x,t)+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(8)

$$\psi (x,t)={\mathcal{Y}}_2\left(\zeta \right)e^{i\left(Z_2(x,t)+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(9)

and

$$\zeta =x-\Im t,Z_i(x,t)=-{\mathit{s}}_{i}x+{\upsilon }_{i}t,i=1,2.$$

(10)

The amplitude portions of the solitons are represented by ${\mathcal{Y}}_i\left(\zeta \right),i=1,2$, whereas the phase components of the solitons are represented by $Z_i(x,t)$, and the frequencies, wave numbers, and velocities of the solitons are represented by ${\mathit{s}}_i,{\upsilon }_i$, and ℑ, respectively.

When compensating from Equations (8)–(10) into Equations (1) and (2), with the separation of the real and imaginary parts, we obtain:

$$\left[{\wp }_{11}-\Im \left(2{\wp }_{14}{\mathit{s}}_1+{\wp }_{12}\right)+3{\wp }_{13}{\mathit{s}}_1+{\wp }_{14}\left({\mathcal{L}}_1^2-{\upsilon }_1\right)\right]{\mathcal{Y}}_1^{″}-{\mathit{s}}_1\left[\left({\beta }_1+{{\rm Y}}_1\right){\mathcal{Y}}_2^2{\mathcal{Y}}_1+\left({\theta }_1+{\lambda }_1\right){\mathcal{Y}}_1^3\right]-$$

$$\left[{\mathit{s}}_1^2\left({\wp }_{13}{\mathit{s}}_1+{\wp }_{11}\right)+\left({\wp }_{14}{\mathit{s}}_1^2+{\wp }_{12}{\mathit{s}}_1-1\right)\left({\mathcal{L}}_1^2-{\upsilon }_1\right)\right]{\mathcal{Y}}_1=0,$$

(11)

$$\left[{\wp }_{21}-\Im \left(2{\wp }_{24}{\mathit{s}}_2+{\wp }_{22}\right)+3{\wp }_{23}{\mathit{s}}_2+{\wp }_{24}\left({\mathcal{L}}_2^2-{\upsilon }_2\right)\right]{\mathcal{Y}}_2^{″}-{\mathit{s}}_2\left[\left({\beta }_2+{{\rm Y}}_2\right){\mathcal{Y}}_1^2{\mathcal{Y}}_2+\left({\theta }_2+{\lambda }_2\right){\mathcal{Y}}_2^3\right]-$$

$$\left[{\mathit{s}}_2^2\left({\wp }_{23}{\mathit{s}}_2+{\wp }_{21}\right)+\left({\wp }_{24}{\mathit{s}}_2^2+{\wp }_{22}{\mathit{s}}_2-1\right)\left({\mathcal{L}}_2^2-{\upsilon }_2\right)\right]{\mathcal{Y}}_2=0,$$

(12)

$$\left({\wp }_{13}-{\wp }_{14}\Im \right){\mathcal{Y}}_1^{\left(3\right)}-\left({\theta }_1+3{\lambda }_1+2{\mu }_1\right){\mathcal{Y}}_1^2{\mathcal{Y}}_1^{\prime }-\left({\beta }_1+{{\rm Y}}_1\right){\mathcal{Y}}_2^2{\mathcal{Y}}_1^{\prime }-2\left({\alpha }_1+{{\rm Y}}_1\right){\mathcal{Y}}_1{\mathcal{Y}}_2{\mathcal{Y}}_2^{\prime }-$$

$$\left[\Im \left(1-{\wp }_{14}{\mathit{s}}_1^2-{\wp }_{12}{\mathit{s}}_1\right)+3{\wp }_{13}{\mathit{s}}_1^2+2{\wp }_{11}{\mathit{s}}_1+\left(2{\wp }_{14}{\mathit{s}}_1+{\wp }_{12}\right)\left({\mathcal{L}}_1^2-{\upsilon }_1\right)\right]{\mathcal{Y}}_1^{\prime }=0,$$

(13)

$$\left({\wp }_{23}-{\wp }_{24}\Im \right){\mathcal{Y}}_2^{\left(3\right)}-\left({\theta }_2+3{\lambda }_2+2{\mu }_2\right){\mathcal{Y}}_2^2{\mathcal{Y}}_2^{\prime }-\left({\beta }_2+{{\rm Y}}_2\right){\mathcal{Y}}_1^2{\mathcal{Y}}_2^{\prime }-2\left({\alpha }_2+{{\rm Y}}_2\right){\mathcal{Y}}_2{\mathcal{Y}}_1{\mathcal{Y}}_1^{\prime }-$$

$$\left[\Im \left(1-{\wp }_{24}{\mathit{s}}_2^2-{\wp }_{22}{\mathit{s}}_2\right)+3{\wp }_{23}{\mathit{s}}_2^2+2{\wp }_{21}{\mathit{s}}_2+\left(2{\wp }_{24}{\mathit{s}}_2+{\wp }_{22}\right)\left({\mathcal{L}}_2^2-{\upsilon }_2\right)\right]{\mathcal{Y}}_2^{\prime }=0.$$

(14)

Set

$${\mathcal{Y}}_2=T{\mathcal{Y}}_1,$$

(15)

where T is a non-zero constant. Then, Equations (11)–(14) convert to

$$\left[{\wp }_{11}-\Im \left(2{\wp }_{14}{\mathit{s}}_1+{\wp }_{12}\right)+3{\wp }_{13}{\mathit{s}}_1+{\wp }_{14}\left({\mathcal{L}}_1^2-{\upsilon }_1\right)\right]{\mathcal{Y}}_1^{″}-{\mathit{s}}_1\left[\left({\beta }_1+{{\rm Y}}_1\right)T^2+\left({\theta }_1+{\lambda }_1\right)\right]{\mathcal{Y}}_1^3-$$

$$\left[{\mathit{s}}_1^2\left({\wp }_{13}{\mathit{s}}_1+{\wp }_{11}\right)+\left({\wp }_{14}{\mathit{s}}_1^2+{\wp }_{12}{\mathit{s}}_1-1\right)\left({\mathcal{L}}_1^2-{\upsilon }_1\right)\right]{\mathcal{Y}}_1=0,$$

(16)

$$\left[{\wp }_{21}-\Im \left(2{\wp }_{24}{\mathit{s}}_2+{\wp }_{22}\right)+3{\wp }_{23}{\mathit{s}}_2+{\wp }_{24}\left({\mathcal{L}}_2^2-{\upsilon }_2\right)\right]T{\mathcal{Y}}_1^{″}-{\mathit{s}}_2\left[\left({\beta }_2+{{\rm Y}}_2\right)+\left({\theta }_2+{\lambda }_2\right)T^2\right]T{\mathcal{Y}}_1^3-$$

$$\left[{\mathit{s}}_2^2\left({\wp }_{23}{\mathit{s}}_2+{\wp }_{21}\right)+\left({\wp }_{24}{\mathit{s}}_2^2+{\wp }_{22}{\mathit{s}}_2-1\right)\left({\mathcal{L}}_2^2-{\upsilon }_2\right)\right]T{\mathcal{Y}}_1=0,$$

(17)

$$\left({\wp }_{13}-{\wp }_{14}\Im \right){\mathcal{Y}}_1^{\left(3\right)}-\left[\left({\theta }_1+3{\lambda }_1+2{\mu }_1\right)+\left[\left({\beta }_1+{{\rm Y}}_1\right)+2\left({\alpha }_1+{{\rm Y}}_1\right)\right]T^2\right]{\mathcal{Y}}_1^2{\mathcal{Y}}_1^{\prime }-$$

$$\left[\Im \left(1-{\wp }_{14}{\mathit{s}}_1^2-{\wp }_{12}{\mathit{s}}_1\right)+3{\wp }_{13}{\mathit{s}}_1^2+2{\wp }_{11}{\mathit{s}}_1+\left(2{\wp }_{14}{\mathit{s}}_1+{\wp }_{12}\right)\left({\mathcal{L}}_1^2-{\upsilon }_1\right)\right]{\mathcal{Y}}_1^{\prime }=0,$$

(18)

$$\left({\wp }_{23}-{\wp }_{24}\Im \right)T{\mathcal{Y}}_1^{\left(3\right)}-\left[\left({\theta }_2+3{\lambda }_2+2{\mu }_2\right)T^2+\left({\beta }_2+{{\rm Y}}_2\right)+2\left({\alpha }_2+{{\rm Y}}_2\right)\right]T{\mathcal{Y}}_1^2{\mathcal{Y}}_1^{\prime }-$$

$$\left[\Im \left(1-{\wp }_{24}{\mathit{s}}_2^2-{\wp }_{22}{\mathit{s}}_2\right)+3{\wp }_{23}{\mathit{s}}_2^2+2{\wp }_{21}{\mathit{s}}_2+\left(2{\wp }_{24}{\mathit{s}}_2+{\wp }_{22}\right)\left({\mathcal{L}}_2^2-{\upsilon }_2\right)\right]T{\mathcal{Y}}_1^{\prime }=0.$$

(19)

By integrating Equations (18) and (19), under the special choice of the integration constant as zero, we obtain:

$$\left({\wp }_{13}-{\wp }_{14}\Im \right){\mathcal{Y}}_1^{″}-\frac{1}{3}\left[\left({\theta }_1+3{\lambda }_1+2{\mu }_1\right)+\left[\left({\beta }_1+{{\rm Y}}_1\right)+2\left({\alpha }_1+{{\rm Y}}_1\right)\right]T^2\right]{\mathcal{Y}}_1^3-$$

$$\left[\Im \left(1-{\wp }_{14}{\mathit{s}}_1^2-{\wp }_{12}{\mathit{s}}_1\right)+3{\wp }_{13}{\mathit{s}}_1^2+2{\wp }_{11}{\mathit{s}}_1+\left(2{\wp }_{14}{\mathit{s}}_1+{\wp }_{12}\right)\left({\mathcal{L}}_1^2-{\upsilon }_1\right)\right]{\mathcal{Y}}_1=0,$$

(20)

$$\left({\wp }_{23}-{\wp }_{24}\Im \right)T{\mathcal{Y}}_1^{″}-\frac{1}{3}\left[\left({\theta }_2+3{\lambda }_2+2{\mu }_2\right)T^2+\left({\beta }_2+{{\rm Y}}_2\right)+2\left({\alpha }_2+{{\rm Y}}_2\right)\right]T{\mathcal{Y}}_1^3-$$

$$\left[\Im \left(1-{\wp }_{24}{\mathit{s}}_2^2-{\wp }_{22}{\mathit{s}}_2\right)+3{\wp }_{23}{\mathit{s}}_2^2+2{\wp }_{21}{\mathit{s}}_2+\left(2{\wp }_{24}{\mathit{s}}_2+{\wp }_{22}\right)\left({\mathcal{L}}_2^2-{\upsilon }_2\right)\right]T{\mathcal{Y}}_1=0.$$

(21)

When applying the linearly independent principle to Equations (20) and (21), we obtain the speed and wave numbers of the solitons as:

$$\Im =\frac{{\wp }_{13}}{{\wp }_{14}}=\frac{{\wp }_{23}}{{\wp }_{24}},$$

(22)

and

$${\upsilon }_i=\frac{2{\wp }_{i1}{\mathit{s}}_i-\Im {\wp }_{i2}{\mathit{s}}_i+{\wp }_{i2}{\mathcal{L}}_i^2+3{\wp }_{i3}{\mathit{s}}_i^2-\Im {\wp }_{i4}{\mathit{s}}_i^2+2{\wp }_{i4}{\mathit{s}}_i{\mathcal{L}}_i^2+\Im }{2{\wp }_{i4}{\mathit{s}}_i+{\wp }_{i2}},$$

(23)

with

$$\left({\theta }_1+3{\lambda }_1+2{\mu }_1\right)+\left[\left({\beta }_1+{{\rm Y}}_1\right)+2\left({\alpha }_1+{{\rm Y}}_1\right)\right]T^2=\left[\left({\theta }_2+3{\lambda }_2+2{\mu }_2\right)T^2+\left({\beta }_2+{{\rm Y}}_2\right)+2\left({\alpha }_2+{{\rm Y}}_2\right)\right]T=0.$$

(24)

Then, Equations (16) and (17) are equivalent under the constraints:

$${\wp }_{11}-\Im \left(2{\wp }_{14}{\mathit{s}}_1+{\wp }_{12}\right)+3{\wp }_{13}{\mathit{s}}_1+{\wp }_{14}\left({\mathcal{L}}_1^2-{\upsilon }_1\right)=\left[{\wp }_{21}-\Im \left(2{\wp }_{24}{\mathit{s}}_2+{\wp }_{22}\right)+3{\wp }_{23}{\mathit{s}}_2+{\wp }_{24}\left({\mathcal{L}}_2^2-{\upsilon }_2\right)\right]T,$$

(25)

$${\mathit{s}}_1\left[\left({\beta }_1+{{\rm Y}}_1\right)T^2+\left({\theta }_1+{\lambda }_1\right)\right]={\mathit{s}}_2\left[\left({\beta }_2+{{\rm Y}}_2\right)+\left({\theta }_2+{\lambda }_2\right)T^2\right]T,$$

(26)

$${\mathit{s}}_1^2\left({\wp }_{13}{\mathit{s}}_1+{\wp }_{11}\right)+\left({\wp }_{14}{\mathit{s}}_1^2+{\wp }_{12}{\mathit{s}}_1-1\right)\left({\mathcal{L}}_1^2-{\upsilon }_1\right)=\left[{\mathit{s}}_2^2\left({\wp }_{23}{\mathit{s}}_2+{\wp }_{21}\right)+\left({\wp }_{24}{\mathit{s}}_2^2+{\wp }_{22}{\mathit{s}}_2-1\right)\left({\mathcal{L}}_2^2-{\upsilon }_2\right)\right]T.$$

(27)

In addition, Equation (16) can be represented as follows:

$${\mathcal{R}}_1{\mathcal{Y}}_1^{″}+{\mathcal{R}}_2{\mathcal{Y}}_1^3+{\mathcal{R}}_3{\mathcal{Y}}_1=0,$$

(28)

where ${\mathcal{R}}_1={\wp }_{11}-\Im \left(2{\wp }_{14}{\mathit{s}}_1+{\wp }_{12}\right)+3{\wp }_{13}{\mathit{s}}_1+{\wp }_{14}\left({\mathcal{L}}_1^2-{\upsilon }_1\right),{\mathcal{R}}_2={\mathit{s}}_1\left[({\beta }_1+{{\rm Y}}_1)T^2+\left({\theta }_1+{\lambda }_1\right)\right],{\mathcal{R}}_3={\mathit{s}}_1^2\left({\wp }_{13}{\mathit{s}}_1+{\wp }_{11}\right)+\left({\wp }_{14}{\mathit{s}}_1^2+{\wp }_{12}{\mathit{s}}_1-1\right)\left({\mathcal{L}}_1^2-{\upsilon }_1\right).$

According to the modified extended mapping method, the solution for Equation (28) can be expressed as follows:

$${\mathcal{Y}}_1\left(\zeta \right)={\mathit{d}}_0+{\mathit{d}}_1\chi \left(\zeta \right)+{\mathit{f}}_1\left(\frac{1}{\chi \left(\zeta \right)}\right)+{\mathit{h}}_1\left(\frac{{\chi }^{\prime }\left(\zeta \right)}{\chi \left(\zeta \right)}\right),$$

(29)

where ${\mathit{d}}_i(i=0,1),{\mathit{f}}_1$ and ${\mathit{h}}_1$ are constants, which we can set under the restrictions ${\mathit{d}}_1$ or ${\mathit{f}}_1$ or ${\mathit{h}}_1\ne 0$.

Insert Equations (29) and (7) into Equation (28) and then collect the coefficients of the same powers and set them equal to zero. This yields a system of algebraic equations that can be solved using Mathematica software. Then, the following cases are obtained:

Case 1:

${\rho }_0={\rho }_1={\rho }_3={\rho }_6=0$.

Set 1: 

${\mathit{d}}_0=0,{\mathit{d}}_1=\pm \frac{\sqrt{-2{\rho }_4{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}},{\mathit{f}}_1=0,{\mathit{h}}_1=0,{\rho }_2=-\frac{{\mathcal{R}}_3}{{\mathcal{R}}_1}.$

Set 2: 

${\mathit{d}}_0=0,{\mathit{d}}_1={\mathit{f}}_1=0,{\mathit{h}}_1=\pm \frac{\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}},{\rho }_2=\frac{{\mathcal{R}}_3}{2{\mathcal{R}}_1}.$

Set 3: 

${\mathit{d}}_0=0,{\mathit{d}}_1=\pm \frac{\sqrt{-2{\rho }_4{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}},{\mathit{f}}_1=0,{\mathit{h}}_1=\pm \frac{\sqrt{-{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}},{\rho }_2=\frac{2{\mathcal{R}}_3}{{\mathcal{R}}_1}.$

Through set 1, the stochastic solutions of (1) and (2) are in the following forms:

(1.1,1) 

If ${\rho }_4<0,{\rho }_2>0\mathrm{and}{\mathcal{R}}_i>0(i=1,2)$, then,

$${\varphi }_{1.1,1}=\pm \frac{\sqrt{2{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}\sec \mathrm{h}\left[(x-\Im t)\sqrt{{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(30)

$${\mathsf{\Psi }}_{1.1,1}=\pm \frac{T\sqrt{2{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}\sec \mathrm{h}\left[(x-\Im t)\sqrt{{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)}.$$

(31)

represent bright solitons.

(1.1,2) 

If ${\rho }_4>0,{\rho }_2<0,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0$, then,

$${\varphi }_{1.1,2}=\pm \frac{\sqrt{2{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}sec\left[(x-\Im t)\sqrt{-{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(32)

$${\mathsf{\Psi }}_{1.1,2}=\pm \frac{T\sqrt{2{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}sec\left[(x-\Im t)\sqrt{-{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(33)

or

$${\varphi }_{1.1,3}=\pm \frac{\sqrt{2{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}csc\left[(x-\Im t)\sqrt{-{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(34)

$${\mathsf{\Psi }}_{1.1,3}=\pm \frac{T\sqrt{2{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}csc\left[(x-\Im t)\sqrt{-{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(35)

represent singular periodic wave solutions.

(1.1,3) 

If ${\rho }_4>0,{\rho }_2=0,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0,$ then,

$${\varphi }_{1.1,4}=\mp \frac{\sqrt{-2{\mathcal{R}}_1}}{(x-\Im t)\sqrt{{\mathcal{R}}_2}}{e}^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(36)

$${\mathsf{\Psi }}_{1.1,4}=\mp \frac{T\sqrt{-2{\mathcal{R}}_1}}{(x-\Im t)\sqrt{{\mathcal{R}}_2}}{e}^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(37)

represent rational solutions.

Through set 2, the stochastic solutions of (1) and (2) are in the following forms:

(1.2,1) 

If ${\rho }_4<0,{\rho }_2>0,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0$, then,

$${\varphi }_{1.2,1}=\mp \frac{\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tanh\left[(x-\Im t)\sqrt{{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(38)

$${\mathsf{\Psi }}_{1.2,1}=\mp \frac{T\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tanh\left[(x-\Im t)\sqrt{{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(39)

represent dark solitons.

(1.2,2) 

If ${\rho }_4>0,{\rho }_2<0,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0$, then,

$${\varphi }_{1.2,2}=\pm \frac{\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tan\left[(x-\Im t)\sqrt{-{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(40)

$${\mathsf{\Psi }}_{1.2,2}=\pm \frac{T\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tan\left[(x-\Im t)\sqrt{-{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(41)

or

$${\varphi }_{1.2,3}=\mp \frac{\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}cot\left[(x-\Im t)\sqrt{-{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(42)

$${\mathsf{\Psi }}_{1.2,3}=\mp \frac{T\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}cot\left[(x-\Im t)\sqrt{-{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(43)

represent periodic wave solutions.

(1.2,3) 

If ${\rho }_4>0,{\rho }_2=0,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0,$ then,

$${\varphi }_{1.2,4}=\mp \frac{\sqrt{-2{\mathcal{R}}_1}}{(x-\Im t)\sqrt{{\mathcal{R}}_2}}{e}^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(44)

$${\mathsf{\Psi }}_{1.2,4}=\mp \frac{\sqrt{-2{\mathcal{R}}_1}}{(x-\Im t)\sqrt{{\mathcal{R}}_2}}{e}^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(45)

represent rational solutions.

Through set 3, if ${\rho }_4<0,{\rho }_2>0,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0$, then,

$${\varphi }_{1.3}=\pm \frac{\sqrt{-{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}}\left(\sqrt{2{\rho }_2}\sec \mathrm{h}\left[(x-\Im t)\sqrt{{\rho }_2}\right]-tanh\left[(x-\Im t)\sqrt{{\rho }_2}\right]\right)e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(46)

$${\mathsf{\Psi }}_{1.3}=\pm \frac{T\sqrt{-{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}}\left(\sqrt{2{\rho }_2}\sec \mathrm{h}\left[(x-\Im t)\sqrt{{\rho }_2}\right]-tanh\left[(x-\Im t)\sqrt{{\rho }_2}\right]\right)e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(47)

represent combo bright–dark solitons.

Case 2:

${\rho }_0=\frac{{\rho }_2^2}{4{\rho }_4},{\rho }_1={\rho }_3={\rho }_6=0$.

Set 1: 

${\mathit{d}}_0={\mathit{f}}_1={\mathit{h}}_1=0,{\mathit{d}}_1=\pm \frac{\sqrt{-2{\rho }_4{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}},{\rho }_2=-\frac{{\mathcal{R}}_3}{{\mathcal{R}}_1}.$

Set 2: 

${\mathit{d}}_0={\mathit{d}}_1={\mathit{f}}_1=0,{\mathit{h}}_1=\pm \frac{\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}},{\rho }_2=\frac{{\mathcal{R}}_3}{2{\mathcal{R}}_1}.$

Set 3: 

${\mathit{d}}_0={\mathit{d}}_1={\mathit{h}}_1=0,{\mathit{f}}_1=\pm \frac{{\mathcal{R}}_3}{\sqrt{-2{\rho }_4{\mathcal{R}}_1{\mathcal{R}}_2}},{\rho }_2=-\frac{{\mathcal{R}}_3}{{\mathcal{R}}_1},$

Set 4: 

${\mathit{d}}_0={\mathit{h}}_1=0,{\mathit{d}}_1=\pm \frac{\sqrt{-2{\rho }_4{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}},{\mathit{f}}_1=\pm \frac{{\mathcal{R}}_3}{4\sqrt{-2{\rho }_4{\mathcal{R}}_1{\mathcal{R}}_2}},{\rho }_2=-\frac{{\mathcal{R}}_3}{4{\mathcal{R}}_1},$

Set 5: 

${\mathit{d}}_0={\mathit{h}}_1=0,{\mathit{d}}_1=\pm \frac{\sqrt{-2{\rho }_4{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}},{\mathit{f}}_1=\pm \frac{{\mathcal{R}}_3}{2\sqrt{-2{\rho }_4{\mathcal{R}}_1{\mathcal{R}}_2}},{\rho }_2=\frac{{\mathcal{R}}_3}{2{\mathcal{R}}_1},$

Set 6: 

${\mathit{d}}_0=0,{\mathit{d}}_1=\pm \frac{\sqrt{-{\rho }_4{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}},{\mathit{f}}_1=\mp \frac{{\mathcal{R}}_3}{2\sqrt{-2{\rho }_4{\mathcal{R}}_1{\mathcal{R}}_2}},{\mathit{h}}_1=\pm \frac{\sqrt{-{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}},{\rho }_2=-\frac{{\mathcal{R}}_3}{{\mathcal{R}}_1},$

Set 7: 

${\mathit{d}}_0=0,{\mathit{d}}_1=\pm \frac{\sqrt{-{\rho }_4{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}},{\mathit{f}}_1=\mp \frac{{\mathcal{R}}_3}{4\sqrt{-2{\rho }_4{\mathcal{R}}_1{\mathcal{R}}_2}},{\mathit{h}}_1=\pm \frac{\sqrt{-{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}},{\rho }_2=\frac{{\mathcal{R}}_3}{2{\mathcal{R}}_1}.$

Through set 1, the stochastic solutions of (1) and (2) are in the following forms:

(2.1,1) 

If ${\rho }_4>0,{\rho }_2<0,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0$, then,

$${\varphi }_{2.1,1}=\pm \frac{\sqrt{{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tanh\left[(x-\Im t)\sqrt{-\frac{{\rho }_2}{2}}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(48)

$${\mathsf{\Psi }}_{2.1,1}=\pm \frac{T\sqrt{{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tanh\left[(x-\Im t)\sqrt{-\frac{{\rho }_2}{2}}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(49)

represent dark solitons.

(2.1,2) 

If ${\rho }_4>0,{\rho }_2>0,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0,$ then,

$${\varphi }_{2.1,2}=\pm \frac{\sqrt{-{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tan\left[(x-\Im t)\sqrt{\frac{{\rho }_2}{2}}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(50)

$${\mathsf{\Psi }}_{2.1,2}=\pm \frac{T\sqrt{-{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tan\left[(x-\Im t)\sqrt{\frac{{\rho }_2}{2}}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(51)

represent periodic solutions.

Through set 2, the solutions of (1) and (2) are in the following forms:

(2.2,1) 

If ${\rho }_4>0,{\rho }_2<0,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0$, then,

$${\varphi }_{2.2,1}=\pm \frac{2\sqrt{{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}\csc \mathrm{h}\left[(x-\Im t)\sqrt{-2{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(52)

$${\mathsf{\Psi }}_{2.2,1}=\pm \frac{2T\sqrt{{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}\csc \mathrm{h}\left[(x-\Im t)\sqrt{-2{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(53)

represent singular solitons.

(2.2,2) 

If ${\rho }_4>0,{\rho }_2>0,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0,$ then,

$${\varphi }_{2.2,2}=\pm \frac{2\sqrt{-{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}csc\left[(x-\Im t)\sqrt{2{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(54)

$${\mathsf{\Psi }}_{2.2,2}=\pm \frac{2T\sqrt{-{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}csc\left[(x-\Im t)\sqrt{2{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(55)

represent periodic solutions.

Through set 3, the solutions of (1) and (2) are in the following forms:

(2.3,1) 

If ${\rho }_4>0,{\rho }_2<0\mathrm{and}{\mathcal{R}}_1{\mathcal{R}}_2<0$, then,

$${\varphi }_{2.3,1}=\pm \frac{{\mathcal{R}}_3}{\sqrt{{\rho }_2{\mathcal{R}}_1{\mathcal{R}}_2}}coth\left[\frac{(x-\Im t)\sqrt{-{\rho }_2}}{2}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(56)

$${\mathsf{\Psi }}_{2.3,1}=\pm \frac{T{\mathcal{R}}_3}{\sqrt{{\rho }_2{\mathcal{R}}_1{\mathcal{R}}_2}}coth\left[\frac{(x-\Im t)\sqrt{-{\rho }_2}}{2}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(57)

represent singular solitons.

(2.3,2) 

If ${\rho }_4>0,{\rho }_2>0\mathrm{and}{\mathcal{R}}_1{\mathcal{R}}_2<0,$ then,

$${\varphi }_{2.3,2}=\pm \frac{{\mathcal{R}}_3}{\sqrt{-{\rho }_2{\mathcal{R}}_1{\mathcal{R}}_2}}cot\left[\frac{(x-\Im t)\sqrt{{\rho }_2}}{2}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(58)

$${\mathsf{\Psi }}_{2.3,2}=\pm \frac{T{\mathcal{R}}_3}{\sqrt{-{\rho }_2{\mathcal{R}}_1{\mathcal{R}}_2}}cot\left[\frac{(x-\Im t)\sqrt{{\rho }_2}}{2}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(59)

represent periodic solutions.

Through set 4, the stochastic solutions of (1) and (2) are in the following forms:

(2.4,1) 

If ${\rho }_4>0,{\rho }_2<0,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0$, then,

$${\varphi }_{2.4,1}=\mp \frac{2\sqrt{{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}\csc \mathrm{h}\left[(x-\Im t)\sqrt{-2{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(60)

$${\mathsf{\Psi }}_{2.4,1}=\mp \frac{2T\sqrt{{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}\csc \mathrm{h}\left[(x-\Im t)\sqrt{-2{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(61)

represent singular solitons.

(2.2,2) 

If ${\rho }_4>0,{\rho }_2>0,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0,$ then,

$${\varphi }_{2.2,2}=\pm \frac{2\sqrt{-{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}csc\left[(x-\Im t)\sqrt{2{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(62)

$${\mathsf{\Psi }}_{2.2,2}=\pm \frac{2T\sqrt{-{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}csc\left[(x-\Im t)\sqrt{2{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(63)

represent periodic solutions.

Through set 5, the solutions of (1) and (2) are in the following forms:

(2.5,1) 

If ${\rho }_4>0,{\rho }_2<0,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0$, then,

$${\varphi }_{2.5,1}=\pm \frac{2\sqrt{{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}coth\left[(x-\Im t)\sqrt{-2{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(64)

$${\mathsf{\Psi }}_{2.5,1}=\pm \frac{2T\sqrt{{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}coth\left[(x-\Im t)\sqrt{-2{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(65)

represent singular solitons.

(2.5,2) 

If ${\rho }_4>0,{\rho }_2>0,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0,$ then,

$${\varphi }_{2.5,2}=\pm \frac{2\sqrt{-{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}cot\left[(x-\Im t)\sqrt{2{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(66)

$${\mathsf{\Psi }}_{2.5,2}=\pm \frac{2T\sqrt{-{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}cot\left[(x-\Im t)\sqrt{2{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(67)

represent periodic solutions.

Through set 6, the stochastic solutions of (1) and (2) are in the following forms:

(2.6,1) 

If ${\rho }_4>0,{\rho }_2<0,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0$, then,

$${\varphi }_{2.6,1}=\mp \frac{\sqrt{{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}coth\left[\frac{(x-\Im t)\sqrt{-{\rho }_2}}{\sqrt{2}}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(68)

$${\mathsf{\Psi }}_{2.6,1}=\mp \frac{T\sqrt{{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}coth\left[\frac{(x-\Im t)\sqrt{-{\rho }_2}}{\sqrt{2}}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(69)

represent singular solitons.

(2.6,2) 

If ${\rho }_4>0,{\rho }_2>0,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0,$ then,

$${\varphi }_{2.6,2}=\pm \frac{\sqrt{-{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tan\left[\frac{(x-\Im t)\sqrt{{\rho }_2}}{\sqrt{2}}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(70)

$${\mathsf{\Psi }}_{2.6,2}=\pm \frac{T\sqrt{-{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tan\left[\frac{(x-\Im t)\sqrt{{\rho }_2}}{\sqrt{2}}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)}.$$

(71)

represent periodic solutions.

Through set 7, if ${\rho }_4>0,{\rho }_2>0,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0$, then,

• $${\varphi }_{2.7}=\pm \frac{2\sqrt{-{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}csc\left[(x-\Im t)\sqrt{2{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

  (72)

  $${\mathsf{\Psi }}_{2.7}=\pm \frac{2T\sqrt{-{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}csc\left[(x-\Im t)\sqrt{2{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

  (73)

  represent periodic solutions.

Case 3:

If, we set ${\rho }_3={\rho }_4={\rho }_6=0$,

${\mathit{d}}_0={\mathit{d}}_1=0,{\mathit{f}}_1=\pm \frac{\sqrt{-{\rho }_0{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}},{\mathit{h}}_1=\pm \frac{\sqrt{-{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}},{\rho }_2=\frac{2{\mathcal{R}}_3}{{\mathcal{R}}_1},$

then, the stochastic solutions of (1) and (2) are in the following forms:

(3.1) 

If ${\rho }_0=0,{\rho }_2>0,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0$, then,

$${\varphi }_{3.1}=\pm \frac{2\sqrt{-{\rho }_2{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}}\left(coth\left[(x-\Im t)\sqrt{{\rho }_2}\right]\sec \mathrm{h}\left[2(x-\Im t)\sqrt{{\rho }_2}\right]\right)e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(74)

$${\mathsf{\Psi }}_{3.1}=\pm \frac{2T\sqrt{-{\rho }_2{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}}\left(coth\left[(x-\Im t)\sqrt{{\rho }_2}\right]\sec \mathrm{h}\left[2(x-\Im t)\sqrt{{\rho }_2}\right]\right)e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(75)

represent combo singular-bright solitons.

(3.2) 

If ${\rho }_0=0,{\rho }_2<0,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0,$ then,

$${\varphi }_{3.2}=\pm \frac{\sqrt{{\rho }_2{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}}\left[\frac{cos\left[(x-\Im t)\sqrt{-{\rho }_2}\right]}{sin\left[(x-\Im t)\sqrt{-{\rho }_2}\right]-1}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(76)

$${\mathsf{\Psi }}_{3.2}=\pm \frac{T\sqrt{{\rho }_2{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}}\left[\frac{cos\left[(x-\Im t)\sqrt{-{\rho }_2}\right]}{sin\left[(x-\Im t)\sqrt{-{\rho }_2}\right]-1}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(77)

represent periodic solutions.

(3.3) 

If ${\rho }_0>0,{\rho }_2<0,{\rho }_1=0,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0,$ then,

$${\varphi }_{3.3}=\pm \frac{\sqrt{{\rho }_2{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}}\left(csc\left[(x-\Im t)\sqrt{-{\rho }_2}\right]+cot\left[(x-\Im t)\sqrt{-{\rho }_2}\right]\right)e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(78)

$${\mathsf{\Psi }}_{3.3}=\pm \frac{T\sqrt{{\rho }_2{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}}\left(csc\left[(x-\Im t)\sqrt{-{\rho }_2}\right]+cot\left[(x-\Im t)\sqrt{-{\rho }_2}\right]\right)e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(79)

represent periodic solutions.

Case 4:

If we set ${\rho }_0={\rho }_1={\rho }_2={\rho }_6=0$, ${\mathit{d}}_0={\mathit{f}}_1=0,{\mathit{d}}_1=\pm \frac{\sqrt{-{\rho }_4{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}},{\mathit{h}}_1=\pm \frac{\sqrt{-{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}},{\mathcal{R}}_3=0.$

Then,

$${\varphi }_4=\pm \frac{\sqrt{-{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}}\left[\frac{2{\rho }_3^2(x-\Im t)-4{\rho }_4\sqrt{{\rho }_4}}{{\rho }_3^2{(x-\Im t)}^2-4{\rho }_4}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(80)

$${\mathsf{\Psi }}_4=\pm \frac{T\sqrt{-{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}}\left[\frac{2{\rho }_3^2(x-\Im t)-4{\rho }_4\sqrt{{\rho }_4}}{{\rho }_3^2{(x-\Im t)}^2-4{\rho }_4}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(81)

represent stochastic rational solutions with ${\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0.$

Case 5:

${\rho }_0={\rho }_1={\rho }_6=0$.

Set 1: 

${\mathit{d}}_0={\mathit{f}}_1=0,{\mathit{d}}_1=\pm \frac{\sqrt{-{\rho }_4{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}},{\mathit{h}}_1=\pm \frac{\sqrt{-{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}},{\rho }_2=\frac{2{\mathcal{R}}_3}{{\mathcal{R}}_1}.$

Set 2: 

${\mathit{f}}_1={\mathit{h}}_1=0,{\mathit{d}}_0=\pm \frac{\sqrt{-{\mathcal{R}}_3}}{\sqrt{{\mathcal{R}}_2}},{\mathit{d}}_1=\pm \frac{\sqrt{-2{\rho }_4{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}},{\rho }_2=\frac{2{\mathcal{R}}_3}{{\mathcal{R}}_1},{\rho }_3=\frac{2\sqrt{2{\rho }_4{\mathcal{R}}_3}}{\sqrt{{\mathcal{R}}_1}}.$

Set 3: 

${\mathit{d}}_0={\mathit{f}}_1={\mathit{h}}_1=0,{\mathit{d}}_1=\pm \frac{\sqrt{-2{\rho }_4{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}},{\rho }_2=-\frac{{\mathcal{R}}_3}{{\mathcal{R}}_1},{\rho }_3=0.$

Set 4: 

${\mathit{d}}_0={\mathit{d}}_1={\mathit{f}}_1=0,{\mathit{h}}_1=\pm \frac{\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}},{\rho }_2=\frac{{\mathcal{R}}_3}{2{\mathcal{R}}_1},{\rho }_3=0.$

Through set 1, the stochastic solutions of (1) and (2) are in the following forms:

(5.1,1) 

If ${\rho }_4>0,{\rho }_2>0,{\rho }_3^2=4{\rho }_2{\rho }_4,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0$, then,

$${\varphi }_{5.1,1}=\mp \frac{\sqrt{-{\rho }_2{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}}tanh\left[\frac{(x-\Im t)\sqrt{{\rho }_2}}{2}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(82)

$${\mathsf{\Psi }}_{5.1,1}=\mp \frac{T\sqrt{-{\rho }_2{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}}tanh\left[\frac{(x-\Im t)\sqrt{{\rho }_2}}{2}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(83)

represent dark solitons.

(5.1,2) 

If ${\rho }_4>0,{\rho }_2>0,{\rho }_3^2=4{\rho }_2{\rho }_4,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0$, then,

$${\varphi }_{5.1,2}=\mp \frac{\sqrt{-{\rho }_2{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}}coth\left[\frac{(x-\Im t)\sqrt{{\rho }_2}}{2}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(84)

$${\mathsf{\Psi }}_{5.1,2}=\mp \frac{T\sqrt{-{\rho }_2{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}}coth\left[\frac{(x-\Im t)\sqrt{{\rho }_2}}{2}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(85)

represent singular solitons.

Through set 2, the solutions for Equations (1) and (2) are obtained as follows:

(5.2,1) 

If ${\rho }_4>0,{\rho }_2>0,{\rho }_3^2=4{\rho }_2{\rho }_4,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0$, then,

$${\varphi }_{5.2,1}=\mp \frac{\sqrt{-{\rho }_2{\mathcal{R}}_1}}{2\sqrt{2{\mathcal{R}}_2}}\left(coth\left[\frac{(x-\Im t)\sqrt{{\rho }_2}}{2}\right]+1-2\sqrt{{\rho }_2}\right)e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(86)

$${\mathsf{\Psi }}_{5.2,1}=\mp \frac{T\sqrt{-{\rho }_2{\mathcal{R}}_1}}{2\sqrt{2{\mathcal{R}}_2}}\left(coth\left[\frac{(x-\Im t)\sqrt{{\rho }_2}}{2}\right]+1-2\sqrt{{\rho }_2}\right)e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(87)

represent singular solitons.

(5.2,2) 

If ${\rho }_4>0,{\rho }_2>0,{\rho }_3^2\ne 4{\rho }_2{\rho }_4,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0$, then

$${\varphi }_{5.2,2}=\pm \frac{\sqrt{-{\rho }_2{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}}\left[1+\frac{2\sqrt{{\rho }_2{\rho }_4}{\sec \mathrm{h}}^2\left[\frac{(x-\Im t)\sqrt{{\rho }_2}}{2}\right]}{-{\rho }_3+2\sqrt{{\rho }_2{\rho }_4}tanh\left[\frac{(x-\Im t)\sqrt{{\rho }_2}}{2}\right]}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(88)

$${\mathsf{\Psi }}_{5.2,2}=\pm \frac{T\sqrt{-{\rho }_2{\mathcal{R}}_1}}{\sqrt{2{\mathcal{R}}_2}}\left[1+\frac{2\sqrt{{\rho }_2{\rho }_4}{\sec \mathrm{h}}^2\left[\frac{(x-\Im t)\sqrt{{\rho }_2}}{2}\right]}{-{\rho }_3+2\sqrt{{\rho }_2{\rho }_4}tanh\left[\frac{(x-\Im t)\sqrt{{\rho }_2}}{2}\right]}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)}.$$

(89)

represent combo bright–dark solitons.

Through set 3, the stochastic solutions of (1) and (2) are in the following forms:

(5.3,1) 

If ${\rho }_4>0,{\rho }_2>0,{\rho }_3^2\ne 4{\rho }_2{\rho }_4,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0$, then,

$${\varphi }_{5.3,1}=\pm \frac{\sqrt{-2{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}\csc \mathrm{h}\left[(x-\Im t)\sqrt{{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(90)

$${\mathsf{\Psi }}_{5.3,1}=\pm \frac{T\sqrt{-2{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}\csc \mathrm{h}\left[(x-\Im t)\sqrt{{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(91)

represent singular solitons.

(5.3,2) 

If ${\rho }_4>0,{\rho }_2<0,{\rho }_3^2\ne 4{\rho }_2{\rho }_4,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0$, then,

$${\varphi }_{5.3,2}=\pm \frac{\sqrt{2{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}csc\left[(x-\Im t)\sqrt{-{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(92)

$${\mathsf{\Psi }}_{5.3,2}=\pm \frac{T\sqrt{2{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}csc\left[(x-\Im t)\sqrt{-{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(93)

represent periodic solutions.

Through set 4, the stochastic solutions of (1) and (2) are in the following forms:

(5.4,1) 

If ${\rho }_4>0,{\rho }_2>0,{\rho }_3^2\ne 4{\rho }_2{\rho }_4,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0$, then,

$${\varphi }_{5.4,1}=\mp \frac{\sqrt{-2{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}coth\left[(x-\Im t)\sqrt{{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(94)

$${\mathsf{\Psi }}_{5.4,1}=\mp \frac{T\sqrt{-2{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}coth\left[(x-\Im t)\sqrt{{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(95)

represent singular solitons.

(5.4,2) 

If ${\rho }_4>0,{\rho }_2<0,{\rho }_3^2\ne 4{\rho }_2{\rho }_4,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0$, then,

$${\varphi }_{5.4,2}=\mp \frac{\sqrt{2{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}cot\left[(x-\Im t)\sqrt{-{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(96)

$${\mathsf{\Psi }}_{5.4,2}=\mp \frac{T\sqrt{2{\rho }_2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}cot\left[(x-\Im t)\sqrt{-{\rho }_2}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(97)

represent periodic solutions.

Case 6:

If we set ${\rho }_2={\rho }_4={\rho }_6=0$,

${\mathit{h}}_1={\mathit{d}}_1=0,{\mathit{d}}_0=\pm \frac{\sqrt{-{\mathcal{R}}_3}}{\sqrt{3{\mathcal{R}}_2}},{\mathit{f}}_1=\mp \frac{\sqrt{-2{\rho }_0{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}},{\rho }_2=-\frac{2\sqrt{2{\rho }_0{\mathcal{R}}_3}}{\sqrt{3{\mathcal{R}}_1}},{\rho }_3=\frac{2\sqrt{2{\mathcal{R}}_3^3}}{3\sqrt{3{\rho }_0{\mathcal{R}}_1^3}}.$

Then,

$${\varphi }_6=\mp \frac{\sqrt{-2{\rho }_0{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}\left[\frac{{\rho }_2}{4}+\frac{1}{\wp \left[\frac{(x-\Im t)\sqrt{{\rho }_3}}{2},\frac{-4{\rho }_1}{{\rho }_3},\frac{-4{\rho }_0}{{\rho }_3}\right]}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(98)

$${\mathsf{\Psi }}_6=\mp \frac{T\sqrt{-2{\rho }_0{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}\left[\frac{{\rho }_2}{4}+\frac{1}{\wp \left[\frac{(x-\Im t)\sqrt{{\rho }_3}}{2},\frac{-4{\rho }_1}{{\rho }_3},\frac{-4{\rho }_0}{{\rho }_3}\right]}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(99)

represent stochastic Weierstrass elliptic doubly periodic solutions with ${\rho }_3,{\mathcal{R}}_2>0,\mathrm{and}{\mathcal{R}}_1<0.$

Case 7:

${\rho }_1={\rho }_3={\rho }_6=0$.

Set 1: 

${\mathit{d}}_0={\mathit{d}}_1={\mathit{f}}_1=0,{\mathit{h}}_1=\pm \frac{\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}},{\rho }_2=\frac{{\mathcal{R}}_3}{2{\mathcal{R}}_1}.$

Set 2: 

${\mathit{d}}_0={\mathit{d}}_1={\mathit{f}}_1=0,{\mathit{h}}_1=\pm \frac{\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}},{\rho }_4=0,{\rho }_2=\frac{{\mathcal{R}}_3}{2{\mathcal{R}}_1}.$

Set 3: 

${\mathit{d}}_0={\mathit{d}}_1={\mathit{f}}_1=0,{\mathit{h}}_1=\pm \frac{\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}},{\rho }_0=0,{\rho }_2=\frac{{\mathcal{R}}_3}{2{\mathcal{R}}_1}.$

Through set 1, the stochastic solutions of (1) and (2) are in the following forms:

(7.1,1) 

If ${\rho }_0=1,{\rho }_2=-\left({\mathcal{M}}^2+1\right),{\rho }_4={\mathcal{M}}^2,{\mathcal{R}}_2>0,{\mathcal{R}}_1<0\mathrm{and}0\le \mathcal{M}\le 1$, then,

$${\varphi }_{7.1,1}=\pm \frac{\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}\left[\frac{(\mathcal{M}-1)\mathrm{nd}(x-\Im t)\mathrm{sd}(x-\Im t)}{\mathrm{cd}(x-\Im t)}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(100)

$${\mathsf{\Psi }}_{7.1,1}=\pm \frac{T\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}\left[\frac{(\mathcal{M}-1)\mathrm{nd}(x-\Im t)\mathrm{sd}(x-\Im t)}{\mathrm{cd}(x-\Im t)}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(101)

or

$${\varphi }_{7.1,2}=\pm \frac{\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}\left[\frac{\mathrm{cn}(x-\Im t)\mathrm{dn}(x-\Im t)}{\mathrm{sn}(x-\Im t)}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(102)

$${\mathsf{\Psi }}_{7.1,2}=\pm \frac{T\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}\left[\frac{\mathrm{cn}(x-\Im t)\mathrm{dn}(x-\Im t)}{\mathrm{sn}(x-\Im t)}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(103)

represent Jacobi elliptic function solutions.

If we take $\mathcal{M}=0$, then,

$${\varphi }_{7.1,3}=\mp \frac{\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tan\left[x-\Im t\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(104)

$${\mathsf{\Psi }}_{7.1,3}=\mp \frac{T\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tan\left[x-\Im t\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(105)

or

$${\varphi }_{7.1,4}=\pm \frac{\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}cot\left[x-\Im t\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(106)

$${\mathsf{\Psi }}_{7.1,4}=\pm \frac{T\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}cot\left[x-\Im t\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(107)

represent periodic solutions.

(7.1,2) 

If ${\rho }_0={\mathcal{M}}^2-1,{\rho }_2=-{\mathcal{M}}^2+2,{\rho }_4=-1,{\mathcal{R}}_2>0,{\mathcal{R}}_1<0\mathrm{and}0\le \mathcal{M}\le 1$, then,

$${\varphi }_{7.1,5}=\mp \frac{\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}\left[\frac{\mathcal{M}\mathrm{cn}(x-\Im t)\mathrm{sn}(x-\Im t)}{\mathrm{dn}(x-\Im t)}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(108)

$${\mathsf{\Psi }}_{7.1,5}=\mp \frac{T\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}\left[\frac{\mathcal{M}\mathrm{cn}(x-\Im t)\mathrm{sn}(x-\Im t)}{\mathrm{dn}(x-\Im t)}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(109)

represent Jacobi elliptic function solutions.

If we take $\mathcal{M}=1$, then,

$${\varphi }_{7.1,6}=\mp \frac{\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tanh\left[x-\Im t\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(110)

$${\mathsf{\Psi }}_{7.1,6}=\mp \frac{T\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tanh\left[x-\Im t\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(111)

represent dark solitons.

(7.1,3) 

If ${\rho }_0=-{\mathcal{M}}^2,{\rho }_2=2{\mathcal{M}}^2-1,{\rho }_4=-{\mathcal{M}}^2+1,{\mathcal{R}}_2>0,{\mathcal{R}}_1<0\mathrm{and}0\le \mathcal{M}\le 1$, then,

$${\varphi }_{7.1,7}=\pm \frac{\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}\left[\frac{\mathrm{dc}(x-\Im t)\mathrm{sn}(x-\Im t)}{\mathrm{nc}(x-\Im t)}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(112)

$${\mathsf{\Psi }}_{7.1,7}=\pm \frac{T\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}\left[\frac{\mathrm{dc}(x-\Im t)\mathrm{sn}(x-\Im t)}{\mathrm{nc}(x-\Im t)}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)}.$$

(113)

If we take $\mathcal{M}=1$, then,

$${\varphi }_{7.1,8}=\pm \frac{\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tanh\left[x-\Im t\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(114)

$${\mathsf{\Psi }}_{7.1,8}=\pm \frac{T\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tanh\left[x-\Im t\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(115)

represent dark solitons.

(7.1,4) 

If ${\rho }_0=-1,{\rho }_2=-{\mathcal{M}}^2+2,{\rho }_4={\mathcal{M}}^2-1,{\mathcal{R}}_2>0,{\mathcal{R}}_1<0\mathrm{and}0\le \mathcal{M}\le 1$, then,

$${\varphi }_{7.1,9}=\pm \frac{\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}\left[\frac{\mathcal{M}\mathrm{cd}(x-\Im t)\mathrm{sd}(x-\Im t)}{\mathrm{nd}(x-\Im t)}\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(116)

$${\mathsf{\Psi }}_{7.1,9}=\pm \frac{T\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}\left[\frac{\mathcal{M}\mathrm{cd}(x-\Im t)\mathrm{sd}(x-\Im t)}{\mathrm{nd}(x-\Im t)}\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(117)

represent Jacobi elliptic function solutions.

Through set 2, the stochastic solutions of (1) and (2) are in the following forms:

(7.2,1) 

If ${\rho }_0=1,{\rho }_2=-\left({\mathcal{M}}^2+1\right),{\rho }_4={\mathcal{M}}^2,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0,$ then,

$${\varphi }_{7.2,1}=\mp \frac{\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tan\left[x-\Im t\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(118)

$${\mathsf{\Psi }}_{7.2,1}=\mp \frac{T\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tan\left[x-\Im t\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(119)

or

$${\varphi }_{7.2,2}=\pm \frac{\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}cot\left[x-\Im t\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(120)

$${\mathsf{\Psi }}_{7.2,2}=\pm \frac{T\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}cot\left[x-\Im t\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(121)

represent periodic solutions.

(7.2,2) 

If ${\rho }_0={\mathcal{M}}^2-1,{\rho }_2=-{\mathcal{M}}^2+2,{\rho }_4=-1,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0$, then,

$${\varphi }_{7.2,3}=\pm \frac{\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tanh\left[x-\Im t\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(122)

$${\mathsf{\Psi }}_{7.2,3}=\pm \frac{T\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tanh\left[x-\Im t\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(123)

represent dark solitons.

Through set 3, the stochastic solutions of (1) and (2) are in the following forms:

(7.3,1) 

If ${\rho }_0=-{\mathcal{M}}^2,{\rho }_2=2{\mathcal{M}}^2-1,{\rho }_4=-{\mathcal{M}}^2+1,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0$ then,

$${\varphi }_{7.3,1}=\pm \frac{\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tan\left[x-\Im t\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(124)

$${\mathsf{\Psi }}_{7.3,1}=\pm \frac{T\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tan\left[x-\Im t\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(125)

represent periodic solutions.

(7.3,2) 

If ${\rho }_0={\mathcal{M}}^2-1,{\rho }_2=-{\mathcal{M}}^2+2,{\rho }_4=-1,{\mathcal{R}}_2>0\mathrm{and}{\mathcal{R}}_1<0$, then,

$${\varphi }_{7.3,2}=\mp \frac{\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tanh\left[x-\Im t\right]e^{i\left(-{\mathit{s}}_{1}x+{\upsilon }_{1}t+{\mathcal{L}}_1{\omega }_1\left(t\right)-{\mathcal{L}}_1^{2}t\right)},$$

(126)

$${\mathsf{\Psi }}_{7.3,2}=\mp \frac{T\sqrt{-2{\mathcal{R}}_1}}{\sqrt{{\mathcal{R}}_2}}tanh\left[x-\Im t\right]e^{i\left(-{\mathit{s}}_{2}x+{\upsilon }_{2}t+{\mathcal{L}}_2{\omega }_2\left(t\right)-{\mathcal{L}}_2^{2}t\right)},$$

(127)

represent dark solitons.

4. Graphic Illustration

In this part, the numerical simulations of some of the solutions are illustrated as follows. Figure 1 represents the bright soliton solution of Equation (30). The values of the selected parameters are ${\wp }_{13}=-1.9,{\wp }_{23}=-1.9,{\wp }_{14}=1.5,{\wp }_{24}=1.5,{\wp }_{12}=1.7,{\wp }_{11}=1.7,{\beta }_1=1.8,{{\rm Y}}_1=1.7,{\theta }_1=1.5,{\lambda }_1=1.8,{\mathit{s}}_2=0.5,{\wp }_{21}=1.6,{\wp }_{22}=1.6,{\mathit{s}}_1=-0.5,{\rho }_2=0.9,{\mathcal{L}}_2={\mathcal{L}}_1=0.5$. Figure 2 represents the singular periodic wave solutions of Equation (32). The values of the selected parameters are ${\wp }_{13}=-1.7,{\wp }_{23}=-1.7,{\wp }_{14}=1.2,{\wp }_{24}=1.2,{\wp }_{12}=1.5,{\wp }_{11}=1.5,{\beta }_1=1.3,{{\rm Y}}_1=1.5,{\theta }_1=1.3,{\lambda }_1=1.6,{\mathit{s}}_2=-1.8,{\wp }_{21}=1.1,{\wp }_{22}=1.2,{\mathit{s}}_1=-1.7,{\rho }_2=-0.9,{\mathcal{L}}_2={\mathcal{L}}_1=1.5$. Figure 3 represents the dark soliton solution of Equation (38). The values of the selected parameters are ${\wp }_{13}=1,{\wp }_{23}=1,{\wp }_{14}=2.5,{\wp }_{24}=2.5,{\wp }_{12}=1.5,{\wp }_{11}=1.5,{\beta }_1=1.8,{{\rm Y}}_1=1.7,{\theta }_1=3.5,{\lambda }_1=2.8,{\mathit{s}}_2=-2.5,{\wp }_{21}=2.6,{\wp }_{22}=2.6,{\mathit{s}}_1=-1.5,{\rho }_2=1,{\mathcal{L}}_2={\mathcal{L}}_1=0.5$. Figure 4 represents the singular soliton solution of Equation (64). The values of the selected parameters are ${\wp }_{13}=-1,{\wp }_{23}=-1,{\wp }_{14}=1.4,{\wp }_{24}=1.4,{\wp }_{12}=1.2,{\wp }_{11}=1.2,{\beta }_1=1.8,{{\rm Y}}_1=1.2,{\theta }_1=0.5,{\lambda }_1=0.3,{\mathit{s}}_2=0.5,{\wp }_{21}=0.9,{\wp }_{22}=0.9,{\mathit{s}}_1=0.5,{\rho }_2=-0.9,{\mathcal{L}}_2={\mathcal{L}}_1=1.5$.

Figure 1. Equation (30): 3D and 2D illustrations.

Figure 2. Equation (32): 3D and 2D illustrations.

Figure 3. Equation (38): 3D and 2D illustrations.

Figure 4. Equation (64): 3D and 2D illustrations.

5. Conclusions

In our paper, the Biswas–Arshed equation with multiplicative white noise has been studied successfully using the modified extended mapping method. Various types of stochastic solutions were obtained such as stochastic bright soliton solutions, stochastic dark soliton solutions, stochastic combo bright–dark soliton solutions, stochastic combo singular-bright soliton solutions, stochastic singular soliton solutions, stochastic periodic solutions, stochastic rational solutions, stochastic Weierstrass elliptic doubly periodic solutions, and stochastic Jacobi elliptic function solutions. The constraints on the parameters were considered to guarantee the existence of the obtained solutions. By comparing our results with the outcomes of [37], it has been recognized that the stochastic dark soliton solutions, stochastic combo bright–dark soliton solutions, stochastic combo singular-bright soliton solutions, periodic solutions, rational solutions, and Weierstrass elliptic doubly periodic solutions have been obtained for the first time in this article. Moreover, 3D plots with contour plots of some solutions were introduced to show their features. Moreover, for the physical illustration of the obtained stochastic solutions, the movements of some of the achieved solutions were described graphically (Figure 1, Figure 2, Figure 3 and Figure 4).