Study of Stochastic–Fractional Drinfel’d–Sokolov–Wilson Equation for M-Shaped Rational, Homoclinic Breather, Periodic and Kink-Cross Rational Solutions

Abstract

We explore stochastic–fractional Drinfel’d–Sokolov–Wilson (SFDSW) equations for some wave solutions such as the cross-kink rational wave solution, periodic cross-rational wave solution and homoclinic breather wave solution. We also examine some M-shaped solutions such as the M-shaped rational solution, M-shaped rational solution with one and two kink waves. We also derive the M-shaped interaction with rogue and kink waves and the M-shaped interaction with periodic and kink waves. This model is used in mathematical physics, surface physics, plasma physics, population dynamics and applied sciences. Moreover, we also show our results graphically in different dimensions. We obtain these solutions under some constraint conditions.

1. Introduction

Numerous branches of nonlinear science including plasma physics, geochemistry, solid-state physics, fluid mechanics [1,2,3,4], optical fibres, nuclear physics and chemical physics have been studied through nonlinear evolution equations (NLEEs) [5,6,7,8]. The travelling-wave solution for NLEEs executes a number of analytical and numerical techniques to get an exact solution for these NLEEs [9,10,11,12,13,14,15]. Recently, a variety of external stimulations including random disturbances have been involved in changing physical systems.

A stochastic differential equation (SDE) is a differential equation that has one or more stochastic processes as its terms, with the solution being another stochastic process. SDEs are used to simulate a variety of phenomena, including stock prices and physical models subject to thermal fluctuating. Consequently, SDEs have emerged and gained a lot of significance in modelling phenomena in atmospheric science, fluid mechanics, oceanography, chemistry, physics and biology [16,17].

The fractional derivative models are used for the accurate modelling of those systems that require an accurate modelling of the damping. The advantages of fractional derivatives are their flexibility and nonlocality. These derivatives can approximate real data with a greater flexibility than classical derivatives because they are of fractional order. Moreover, they consider nonlocality, which classical derivatives are unable to achieve. However, a number of significant phenomena such as anomalous diffusion, electrochemistry, acoustics, image processing and electromagnetism are represented by fractional derivative. Fractional models are more precise than integer models. In general, it is more challenging to obtain an exact solution of SDEs with fractional derivatives than classical ones. As a result, we considered the following SFDSW equations given as [18]:

$$du+\left[{\beta }_{1}v{D}_x^{\omega }v\right]dt=\zeta ud\eta ,$$

(1)

$$\begin{array}{c}\hfill dv+[{\beta }_2{D}_{xxx}^{\omega }v+{\beta }_{3}u{D}_x^{\omega }v+{\beta }_{4}v{D}_x^{\omega }u]dt=\zeta vd\eta ,\end{array}$$

(2)

where $u=u(x,t)$, $v=v(x,t)$ and ${\beta }_j$ for $j=1,2,3,4$ are nonzero constant. $\eta =\eta \left(t\right)$ is the standard Brownian motion, $\zeta$ is noise strength, and $D^{\omega }$ is a conformable derivative for $0<\omega <1.$

The remaining manuscript is arranged as follows: In Section 2, we explain the properties and definitions of standard Brownian motion and also discuss Hirota’s bilinear method. In Section 3, we obtain the wave equation for the SFDSW equation. In Section 4 and Section 5, we introduce the solution for the cross-kink rational solution and periodic cross-rational solution, respectively; we also examine the homoclinic breather in Section 6, M-shaped rational wave solution in Section 7, M-shaped rational wave solution with one kink and two kink waves in Section 8 and Section 9, respectively. Moreover, we obtain the M-shaped rational interaction with rogue and kink waves and the M-shaped rational interaction with periodic and kink waves in Section 10 and Section 11. In Section 12, we address results and discussion. Section 13 presents the conclusion of the paper.

2. Preliminaries

Now, we discuss the properties and definitions of a conformable derivative and standard Brownian motion. The definition of a conformable derivative is given as:

Definition 1

([19]). The conformable derivative with order ω of $Q:R^{+}\to R$ is given as:

$$\begin{array}{c}\hfill D_x^{\omega }Q\left(z\right)=\underset{m\to 0}{lim}\frac{Q(z+m{z}^{1-\omega })-Q\left(z\right)}{m}.\end{array}$$

Theorem 1

([19]). Suppose that $Q_1,Q_2:R^{+}\to R$ are ω differential functions,

$$\begin{array}{c}\hfill D_x^{\omega }\left(Q_1\mathrm{o}{Q}_2\right)\left(z\right)=z^{1-\omega }{Q}_2^{\prime }\left(x\right)Q_1\left(Q_2\left(x\right)\right).\end{array}$$

Some properties of the conformable derivative are given as:

1.

$D_z^{\omega }[n_1{Q}_1\left(z\right)+n_2{Q}_2\left(z\right)]=n_1{D}_x^{\omega }{Q}_1\left(z\right)+n_2{D}_x^{\omega }{Q}_2\left(z\right),n_1,n_2ϵR;$

2.

$D_z^{\omega }\left[z^m\right]=m{z}^{m-\omega },mϵR;$

3.

$D_z^{\omega }Q\left(x\right)=z^{1-\omega }\frac{dQ}{dx};$

4.

$D_z^{\omega }\left[K\right]=0$, K is constant.

Definition 2

([20]). A stochastic system ${\left\{\eta \left(t\right)\right\}}_{t\ge 0}$ is a standard Brownian motion if

1.

$\eta \left(0\right)=0$,

2.

$\eta \left(t\right),t\ge 0$, is a continuous function of t;

3.

$\eta \left(t_1\right)-\eta \left(t_2\right)$ is independent for $t_1<t_2$;

4.

With the variance $t_2-t_1$ and mean 0, $\eta \left(t_2\right)-\eta \left(t_1\right)$ has a normal distribution.

Lemma 1

([20]). $E\left(e^{\gamma \eta \left(t\right)}\right)=e^{\frac{1}{2}{\gamma }^{2}t}$ for $\gamma \ge 0.$

Hirota’s Bilinear Method

Hirota invented a method in 1971 to obtain multisoliton solutions of integrable nonlinear evolution equations. A particularly simple manifestation of multisoliton solutions was desired, therefore the aim was to convert existing variables into new ones. Hirota’s method was the quickest to provide results to find soliton solutions [21].

The standard definition of Hirota’s bilinear operators was first introduced by Hirota as:

$$\begin{array}{c}\hfill D_t^n{D}_x^m(\alpha .\beta )={(\frac{\partial }{\partial t}-\frac{\partial }{\partial t^{\prime }})}^n{(\frac{\partial }{\partial x}-\frac{\partial }{\partial x^{\prime }})}^m\alpha (x,t)\beta (x^{\prime },t^{\prime })\mid x^{\prime }=x,t^{\prime }=t^{\prime }.\end{array}$$

This type of equations can typically be made bilinear by including a new dependent variable, such as $logf$ or $\frac{f}{g}$.

3. Wave Transformation for SFDSW

For SFDSW Equations (1) and (2), we use the following wave transformation to build the wave equation [18]:

$$\begin{array}{c}\hfill u(x,t)=U\left(\nu \right)e^{(\zeta \eta \left(t\right)-\frac{1}{2}{\zeta }^{2}t)},v(x,t)=V\left(\nu \right)e^{(\zeta \eta \left(t\right)-\frac{1}{2}{\zeta }^{2}t)},\nu =\frac{1}{\omega }{x}^{\omega }+\varphi t,\end{array}$$

(3)

where U and V are real functions. Inserting Equation (3) into Equations (1) and (2), we have

$$\begin{array}{c}du=[\varphi U^{\prime }dt+\zeta Ud\eta ]e^{(\zeta \eta \left(t\right)-\frac{1}{2}{\zeta }^{2}t)},\\ dv=[\varphi V^{\prime }dt+\zeta Vd\eta ]e^{(\zeta \eta \left(t\right)-\frac{1}{2}{\zeta }^{2}t)},\\ D_x^{\omega }v=V^{\prime }{e}^{(\zeta \eta \left(t\right)-\frac{1}{2}{\zeta }^{2}t)},\\ D_x^{\omega }u=U^{\prime }{e}^{(\zeta \eta \left(t\right)-\frac{1}{2}{\zeta }^{2}t)},\\ D_{xxx}^{\omega }v=V^{\prime ″}{e}^{(\zeta \eta \left(t\right)-\frac{1}{2}{\zeta }^{2}t)}.\end{array}$$

(4)

By using Equation (4) into Equations (1) and (2), we get

$$\begin{array}{c}\hfill \varphi U^{\prime }+{\beta }_{1}V{V}^{\prime }{e}^{(\zeta \eta \left(t\right)-\frac{1}{2}{\zeta }^{2}t)}=0,\end{array}$$

(5)

$$\begin{array}{c}\hfill \varphi V^{\prime }+{\beta }_2{V}^{\prime ″}+{\beta }_{3}U{V}^{\prime }{e}^{(\zeta \eta \left(t\right)-\frac{1}{2}{\zeta }^{2}t)}+{\beta }_{4}V{U}^{\prime }{e}^{(\zeta \eta \left(t\right)-\frac{1}{2}{\zeta }^{2}t)}=0,\end{array}$$

(6)

and we have

$$\begin{array}{c}\hfill \varphi U^{\prime }+{\beta }_{1}V{V}^{\prime }{e}^{-\frac{1}{2}{\zeta }^{2}t}E\left(e^{\zeta }\eta \left(t\right)\right)=0,\end{array}$$

(7)

$$\begin{array}{c}\hfill \varphi V^{\prime }+{\beta }_2{V}^{\prime \prime \prime }+[{\beta }_{3}U{V}^{\prime }+{\beta }_{4}V{U}^{\prime }]e^{-\frac{1}{2}{\zeta }^{2}t}E\left(e^{\zeta }\eta \left(t\right)\right)=0.\end{array}$$

(8)

Using Lemma 1, we have

$$\begin{array}{c}\hfill \varphi U^{\prime }+{\beta }_{1}V{V}^{\prime }=0,\end{array}$$

(9)

$$\begin{array}{c}\hfill \varphi V^{\prime }+{\beta }_2{V}^{\prime ″}+{\beta }_{3}U{V}^{\prime }+{\beta }_{4}V{U}^{\prime }=0.\end{array}$$

(10)

Integrating Equation (9), we obtain

$$\begin{array}{c}\hfill U=-\frac{{\beta }_1}{\varphi }{V}^2+C,\end{array}$$

(11)

where C is a constant. Inserting Equation (11) into Equation (10), and utilizing Equation (9), we obtain

$$\begin{array}{c}\hfill {\beta }_2{V}^{\prime ″}-[\frac{{\beta }_1{\beta }_3}{2\varphi }+\frac{{\beta }_1{\beta }_4}{\varphi }]V^2{V}^{\prime }+[\varphi +C{\beta }_3]V^{\prime }=0.\end{array}$$

(12)

Integrating Equation (12), we have the following wave equation

$$\begin{array}{c}\hfill V^{″}-h_1{V}^3+h_{2}V=0,\end{array}$$

(13)

where $h_1=\frac{{\beta }_1{\beta }_3}{6{\beta }_2\varphi }+\frac{{\beta }_1{\beta }_4}{3{\beta }_2\varphi }$ and $h_2=\frac{\varphi }{{\beta }_2}+\frac{C{\beta }_3}{{\beta }_2}.$ To find the bilinear form of Equation (13), we substitute the following transformation for various solutions [22]:

$$\begin{array}{cc}& V=2{(logf)}_{\nu },\end{array}$$

(14)

$$\begin{array}{cc}& 6{\beta }_{3}C\varphi f^2{f}^{\prime }+6{\beta }_2\varphi f^2{f}^{\left(3\right)}+12{\beta }_2\varphi f^{\prime 3}-4{\beta }_1{\beta }_3{f}^{\prime 3}-8{\beta }_1{\beta }_4{f}^{\prime 3}+6{\varphi }^2{f}^2{f}^{\prime }-18{\beta }_2\varphi f{f}^{\prime }{f}^{″}.\end{array}$$

(15)

Now, we study the following wave solution by using Equation (15):

4. Cross-Kink Rational Wave Solution

We use the following ansatz for the cross-kink rational wave [23]:

$$\begin{array}{cc}& f=e^{-G_1}+r_1{e}^{G_1}+c_1^2+c_2^2+w_5,\\ \hfill & c_1=w_1\nu +w_2,c_2=w_3\nu +w_4,\\ \hfill & G_1=l_1\nu +l_2.\end{array}$$

(16)

Substitute Equation (16) into Equation (15). By equating the coefficients of $x,t{e}^{-3l_1\nu -3l_2},$$e^{-2l_1\nu -2l_2},e^{-l_1\nu -l_2},e^{l_1\nu +l_2},e^{2l_1\nu +2l_2},e^{3l_1\nu +3l_2},e^{2(l_1\nu +l_2)-l_1\nu -l_2},e^{2(l_1\nu +l_2)+l_1\nu +l_2}$ and $e^{2(l_1\nu +l_2)-2l_1\nu -2l_2}$ to zero, we have some values for the wave solution:

$$\begin{array}{c}\hfill l_1=0,w_1=-\frac{w_3{w}_4}{w_2},\\ \hfill w_5=-\frac{{\beta }_{3}C{w}_2^4+{\beta }_{3}C{w}_2^2{w}_4^2+w_2^4\varphi -3{\beta }_2{w}_2^2{w}_3^2+w_2^2{w}_4^2\varphi -3{\beta }_2{w}_3^2{w}_4^2}{w_2^2({\beta }_{3}C+\varphi )}.\end{array}$$

(17)

Putting Equation (17) into Equation (16) and by using them Equation (14), we obtain

$$\begin{array}{c}\hfill V=\frac{2\left(2w_3(\nu w_3+w_4)-\frac{2w_3{w}_4\left(w_2-\frac{\nu w_3{w}_4}{w_2}\right)}{w_2}\right)}{\mathrm{\Xi }+e^{l_2}{r}_1+e^{-l_2}+{\left(w_2-\frac{\nu w_3{w}_4}{w_2}\right)}^2+{(\nu w_3+w_4)}^2}.\end{array}$$

(18)

Putting Equation (18) into Equation (11) yields

$$\begin{array}{c}\hfill U=-\frac{4{\beta }_{1}C{\left(2w_3(\nu w_3+w_4)-\frac{2w_3{w}_4\left(w_2-\frac{\nu w_3{w}_4}{w_2}\right)}{w_2}\right)}^2}{\varphi {\left(\mathrm{\Xi }+e^{l_2}{r}_1+e^{-l_2}+{\left(w_2-\frac{\nu w_3{w}_4}{w_2}\right)}^2+{(\nu w_3+w_4)}^2\right)}^2}.\end{array}$$

(19)

Inserting Equations (18) and (19) into Equation (3), we have

$$\begin{array}{c}\hfill u(x,t)=-\frac{4{\beta }_{1}C{e}^{\zeta \eta \left(t\right)-\frac{{\zeta }^{2}t}{2}}{\left(2w_3\left(\Delta \right)-\frac{2w_3{w}_4\left(\mathsf{\Theta }\right)}{w_2}\right)}^2}{\varphi {\left(\mathrm{\Xi }+e^{l_2}{r}_1+e^{-l_2}+{\left(\mathsf{\Theta }\right)}^2+{\left(w_3\left(t\varphi +\frac{x^{\omega }}{\omega }\right)+w_4\right)}^2\right)}^2},\end{array}$$

(20)

$$\begin{array}{c}\hfill v(x,t)=\frac{2e^{\zeta \eta \left(t\right)-\frac{{\zeta }^{2}t}{2}}\left(2w_3\left(\Delta \right)-\frac{2w_3{w}_4\left(\mathsf{\Theta }\right)}{w_2}\right)}{\mathrm{\Xi }+e^{l_2}{r}_1+e^{-l_2}+{\left(\mathsf{\Theta }\right)}^2+{\left(\Delta \right)}^2},\end{array}$$

(21)

where $\Delta =w_3\left(t\varphi +\frac{x^{\omega }}{\omega }\right)+w_4$, $\mathsf{\Theta }=w_2-\frac{w_3{w}_4\left(t\varphi +\frac{x^{\omega }}{\omega }\right)}{w_2}$$\mathrm{\Xi }=$ $\frac{{\beta }_3(-C)w_2^4-{\beta }_{3}C{w}_2^2{w}_4^2-w_2^4\varphi +3{\beta }_2{w}_2^2{w}_3^2-w_2^2{w}_4^2\varphi +3{\beta }_2{w}_3^2{w}_4^2}{w_2^2({\beta }_{3}C+\varphi )}$.

5. Periodic Cross-Rational Wave Solution

For periodic cross-rational waves, we utilize the given ansatz [24,25]:

$$\begin{array}{cc}& f=c_1^2+c_2^2+r_{1}cos\left(G_1\right)+r_{2}cosh\left(G_2\right)+w_5,\\ \hfill & a_1=w_1\nu +w_2,a_2=w_3\nu +w_4,\\ \hfill & G_1=l_1\nu +l_2,G_2=l_3\nu +l_4.\end{array}$$

(22)

Using Equation (22) into Equation (15) and zeroing the coefficients of $x,tcos(l_1\nu +l_2),$ ${cos}^2(l_1\nu +l_2),cosh(l_3\nu +l_4),cos(l_1\nu +l_2)cosh(l_3\nu +l_4),{cosh}^2(l_3\nu +l_4),sin(l_1\nu +l_2),$ $sin(l_1\nu +l_2)cos(l_1\nu +l_2),sin(l_1\nu +l_2)cosh(l_3\nu +l_4),sin(l_1\nu +l_2)cos(l_1\nu +l_2)cosh(l_3\nu +l_4),sin(l_1\nu +l_2){cosh}^2(l_3\nu +l_4),sinh(l_3\nu +l_4),cos(l_1\nu +l_2)sinh(l_3\nu +l_4),$$sinh(l_3\nu +l_4)cosh(l_3\nu +l_4),cos(l_1\nu +l_2)sinh(l_3\nu +l_4)cosh(l_3\nu +l_4),$$sinh(l_3\nu +l_4){cosh}^2(l_3\nu +l_4)$ and ${sinh}^3(l_3\nu +l_4)$, we get some values for the periodic cross-rational wave solution:

$$\begin{array}{c}\hfill l_3=\sqrt{-\frac{{\beta }_{3}C+\varphi }{{\beta }_2}},w_1=-\frac{w_3{w}_4}{w_2},\end{array}$$

(23)

Putting Equation (23) into Equation (22) and then inserting into Equation (14), we obtain

$$\begin{array}{c}\hfill V=\frac{2\left(r_2\sqrt{\frac{{\beta }_3(-C)-\varphi }{{\beta }_2}}sinh\left(\phi \right)-l_1{r}_{1}sin(l_1\nu +l_2)-\frac{2w_3{w}_4\left(w_2-\frac{\nu w_3{w}_4}{w_2}\right)}{w_2}+2w_3(\nu w_3+w_4)\right)}{r_{2}cosh\left(\phi \right)+r_{1}cos(l_1\nu +l_2)+{\left(w_2-\frac{\nu w_3{w}_4}{w_2}\right)}^2+{(\nu w_3+w_4)}^2+w_5}.\end{array}$$

(24)

Inserting Equation (24) into Equation (11) yields

$$\begin{array}{c}\hfill U=-\frac{4{\beta }_{1}C{\left(r_2\sqrt{\frac{{\beta }_3(-C)-\varphi }{{\beta }_2}}sinh\left(\phi \right)-l_1{r}_{1}sin(l_1\nu +l_2)-\frac{2w_3{w}_4\left(w_2-\frac{\nu w_3{w}_4}{w_2}\right)}{w_2}+2w_3(\nu w_3+w_4)\right)}^2}{\varphi {\left(r_{2}cosh\left(\phi \right)+r_{1}cos(l_1\nu +l_2)+{\left(w_2-\frac{\nu w_3{w}_4}{w_2}\right)}^2+{(\nu w_3+w_4)}^2+w_5\right)}^2},\end{array}$$

(25)

where $\phi =\nu \sqrt{\frac{{\beta }_3(-C)-\varphi }{{\beta }_2}}+l_4$.

Substituting Equations (24) and (25) into Equation (3), we get

$$\begin{array}{c}\hfill u(x,t)=-\frac{4{\beta }_{1}C{e}^{\zeta \eta \left(t\right)-\frac{{\zeta }^{2}t}{2}}{\left(r_2\sqrt{\frac{{\beta }_3(-C)-\varphi }{{\beta }_2}}sinh\left({\mathsf{\Theta }}_1\right)-l_1{r}_{1}sin\left({\mathsf{\Theta }}_2\right)-\frac{2w_3{w}_4\left(\mathsf{\Lambda }\right)}{w_2}+2w_3\left(\psi \right)\right)}^2}{\varphi {\left(r_{2}cosh\left({\mathsf{\Theta }}_1\right)+r_{1}cos\left({\mathsf{\Theta }}_2\right)+{\left(\mathsf{\Lambda }\right)}^2+{\left(\psi \right)}^2+w_5\right)}^2},\end{array}$$

(26)

$$\begin{array}{c}\hfill v(x,t)=\frac{2e^{\zeta \eta \left(t\right)-\frac{{\zeta }^{2}t}{2}}\left(r_2\sqrt{\frac{{\beta }_3(-C)-\varphi }{{\beta }_2}}sinh\left({\mathsf{\Theta }}_1\right)-l_1{r}_{1}sin\left({\mathsf{\Theta }}_2\right)-\frac{2w_3{w}_4\left(\mathsf{\Lambda }\right)}{w_2}+2w_3\left(\psi \right)\right)}{r_{2}cosh\left({\mathsf{\Theta }}_1\right)+r_{1}cos\left({\mathsf{\Theta }}_2\right)+{\left(\mathsf{\Lambda }\right)}^2+{\left(\psi \right)}^2+w_5},\end{array}$$

(27)

where ${\mathsf{\Theta }}_1=\sqrt{\frac{{\beta }_3(-C)-\varphi }{{\beta }_2}}\left(t\varphi +\frac{x^{\omega }}{\omega }\right)+l_4$, ${\mathsf{\Theta }}_2=l_1\left(t\varphi +\frac{x^{\omega }}{\omega }\right)+l_2$, $\mathsf{\Lambda }=w_2-\frac{w_3{w}_4\left(t\varphi +\frac{x^{\omega }}{\omega }\right)}{w_2}$ and $\psi =w_3\left(t\varphi +\frac{x^{\omega }}{\omega }\right)+w_4.$

6. Homoclinic Breather Wave Solution

For homoclinic breather pulses, we use the following ansatz [26,27]:

$$\begin{array}{cc}& f=e^{-w(l_1\nu +l_2)}+r_1{e}^{l(w_3\nu +w_4)}+r_{2}cos\left(w_1(l_5\nu +l_6)\right).\end{array}$$

(28)

Putting Equation (28) into Equation (15), and setting the coefficients of $x,t,e^{-3w(l_1\nu +l_2)},$ $e^{w(l_3\nu +l_4)-2w(l_1\nu +l_2)},e^{2w(l_3\nu +l_4)-w(l_1\nu +l_2)},e^{3w(l_3\nu +l_4)},e^{-2w(l_1\nu +l_2)}cos\left(w_1(l_5\nu +l_6)\right),cos\left(w_1(l_5\nu +l_6)\right)e^{w(l_3\nu +l_4)-w(l_1\nu +l_2)},e^{2w(l_3\nu +l_4)}cos\left(w_1(l_5\nu +l_6)\right),e^{-w(l_1\nu +l_2)}$${cos}^2\left(w_1(l_5\nu +l_6)\right),6e^{w(l_3\nu +l_4)}{cos}^2\left(w_1(l_5\nu +l_6)\right),e^{-2w(l_1\nu +l_2)}sin\left(w_1(l_5\nu +l_6)\right),$$sin\left(w_1(l_5\nu +l_6)\right)e^{w(l_3\nu +l_4)-w(l_1\nu +l_2)},e^{2w(l_3\nu +l_4)}sin\left(w_1(l_5\nu +l_6)\right),$$e^{-w(l_1\nu +l_2)}sin\left(w_1(l_5\nu +l_6)\right)cos\left(w_1(l_5\nu +l_6)\right),e^{w(l_3\nu +l_4)}sin\left(w_1(l_5\nu +l_6)\right)cos\left(w_1(l_5\nu +l_6)\right),$$e^{-2w(l_1\nu +l_2)}cos\left(w_1(l_5\nu +l_6)\right),cos\left(w_1(l_5\nu +l_6)\right)e^{w(l_3\nu +l_4)-w(l_1\nu +l_2)},$$cos\left(w_1(l_5\nu +l_6)\right)e^{w(l_3\nu +l_4)-w(l_1\nu +l_2)},sin\left(w_1(l_5\nu +l_6)\right){cos}^2\left(w_1(l_5\nu +l_6)\right),$$e^{-w(l_1\nu +l_2)}{sin}^2\left(w_1(l_5\nu +l_6)\right)$ and $e^{w(l_3\nu +l_4)}{sin}^2\left(w_1(l_5\nu +l_6)\right)$ to zero, we obtain the following for the homoclinic breather wave solution:

$$\begin{array}{c}\hfill l_3=0,l_1=\frac{\sqrt{-\frac{4{\beta }_{3}C+4\varphi }{{\beta }_2}}}{w},l_5=\frac{\sqrt{-\frac{2{\beta }_{3}C+2\varphi }{{\beta }_2}}}{w_1}.\end{array}$$

(29)

Using Equation (29) into Equation (28) and then using into Equation (14), we have

$$\begin{array}{c}\hfill V=\frac{2\left(-\sqrt{\frac{-4{\beta }_{3}C-4\varphi }{{\beta }_2}}{e}^{-w\left(\frac{\nu \sqrt{\frac{-4{\beta }_{3}C-4\varphi }{{\beta }_2}}}{w}+l_2\right)}-r_2\sqrt{\frac{-2{\beta }_{3}C-2\varphi }{{\beta }_2}}sin\left(w_1\left(\frac{\nu \sqrt{\frac{-2{\beta }_{3}C-2\varphi }{{\beta }_2}}}{w_1}+l_6\right)\right)\right)}{e^{-w\left(\frac{\nu \sqrt{\frac{-4{\beta }_{3}C-4\varphi }{{\beta }_2}}}{w}+l_2\right)}+r_{2}cos\left(w_1\left(\frac{\nu \sqrt{\frac{-2{\beta }_{3}C-2\varphi }{{\beta }_2}}}{w_1}+l_6\right)\right)+r_1{e}^{l_{4}w}}.\end{array}$$

(30)

Inserting Equation (30) into Equation (11) yields

$$\begin{array}{c}\hfill U=-\frac{4\beta 1C{\left(-\sqrt{\frac{-4{\beta }_{3}C-4\varphi }{{\beta }_2}}{e}^{-w\left(\frac{\nu \sqrt{\frac{-4{\beta }_{3}C-4\varphi }{{\beta }_2}}}{w}+l_2\right)}-r_2\sqrt{\frac{-2{\beta }_{3}C-2\varphi }{{\beta }_2}}sin\left(w_1\left(\frac{\nu \sqrt{\frac{-2{\beta }_{3}C-2\varphi }{{\beta }_2}}}{w_1}+l_6\right)\right)\right)}^2}{\varphi {\left(e^{-w\left(\frac{\nu \sqrt{\frac{-4{\beta }_{3}C-4\varphi }{{\beta }_2}}}{w}+l_2\right)}+r_{2}cos\left(w_1\left(\frac{\nu \sqrt{\frac{-2{\beta }_{3}C-2\varphi }{{\beta }_2}}}{w_1}+l_6\right)\right)+r_1{e}^{l_{4}w}\right)}^2}.\end{array}$$

(31)

Inserting Equation (30) into Equation (31) and then into Equation (3), we get the solution for Y and $\mathsf{\Psi }$,

$$\begin{array}{c}\hfill u(x,t)=e^{\zeta \eta \left(t\right)-\frac{{\zeta }^{2}t}{2}}\left(-\frac{4{\beta }_{1}C{\left(-\sqrt{\frac{-4{\beta }_{3}C-4\varphi }{{\beta }_2}}{e}^{\lambda }-r_2\sqrt{\frac{-2{\beta }_{3}C-2\varphi }{{\beta }_2}}sin\left(\mathsf{\Omega }\right)\right)}^2}{\varphi {\left(e^{\lambda }+r_{2}cos\left(\mathsf{\Omega }\right)+r_1{e}^{l_{4}w}\right)}^2}\right)\left(t\varphi +\frac{x^{\omega }}{\omega }\right),\end{array}$$

(32)

$$\begin{array}{c}\hfill v(x,t)=\frac{2e^{\zeta \eta \left(t\right)-\frac{{\zeta }^{2}t}{2}}\left(-\sqrt{\frac{-4{\beta }_{3}C-4\varphi }{{\beta }_2}}{e}^{\lambda }-r_2\sqrt{\frac{-2{\beta }_{3}C-2\varphi }{{\beta }_2}}sin\left(\mathsf{\Omega }\right)\right)}{e^{\lambda }+r_{2}cos\left(\mathsf{\Omega }\right)+r_1{e}^{l_{4}w}},\end{array}$$

(33)

where $\mathsf{\Omega }=w_1\left(\frac{\sqrt{\frac{-2{\beta }_{3}C-2\varphi }{{\beta }_2}}\left(t\varphi +\frac{x^{\omega }}{\omega }\right)}{w_1}+l_6\right)$ and $\lambda =-w\left(\frac{\sqrt{\frac{-4{\beta }_{3}C-4\varphi }{{\beta }_2}}\left(t\varphi +\frac{x^{\omega }}{\omega }\right)}{w}+l_2\right)$.

7. M-Shaped Rational Wave Solution

For the M-shaped rational wave solution, we use the following transformation [28]:

$$\begin{array}{cc}& f=c_1^2+c_2^2+w_5,\\ \hfill & c_1=w_1\nu +w_2,c_2=w_3\nu +w_4.\end{array}$$

(34)

Using Equation (34) into Equation (15), we get the following values for the solution:

$$\begin{array}{c}\hfill w_1=0,w_4=0,w_5=-\frac{3{\beta }_{3}C{w}_2^2\varphi +3w_2^2{\varphi }^2-4{\beta }_1{\beta }_3{w}_3^2-8{\beta }_1{\beta }_4{w}_3^2+3{\beta }_2{w}_3^2\varphi }{3\left(\varphi ({\beta }_{3}C+\varphi )\right)}.\end{array}$$

(35)

Using Equation (35) into Equation (34) and then inserting into Equation (14), we get

$$\begin{array}{c}\hfill V=\frac{4\nu w_3^2}{-\frac{3{\beta }_{3}C{w}_2^2\varphi +3w_2^2{\varphi }^2-4{\beta }_1{\beta }_3{w}_3^2-8{\beta }_1{\beta }_4{w}_3^2+3{\beta }_2{w}_3^2\varphi }{3\varphi ({\beta }_{3}C+\varphi )}+w_2^2+{\nu }^2{w}_3^2}.\end{array}$$

(36)

By using Equation (36) into Equation (11), we have

$$\begin{array}{c}\hfill U=-\frac{16{\beta }_{1}C{\nu }^2{w}_3^4}{\varphi {\left(-\frac{3{\beta }_{3}C{w}_2^2\varphi +3w_2^2{\varphi }^2-4{\beta }_1{\beta }_3{w}_3^2-8{\beta }_1{\beta }_4{w}_3^2+3{\beta }_2{w}_3^2\varphi }{3\varphi ({\beta }_{3}C+\varphi )}+w_2^2+{\nu }^2{w}_3^2\right)}^2}.\end{array}$$

(37)

Substituting Equation (36) into Equation (37) and then into Equation (3), we have the solution for Y and $\mathsf{\Psi }$,

$$\begin{array}{c}\hfill u(x,t)=-\frac{16{\beta }_{1}C{w}_3^4{e}^{\zeta \eta \left(t\right)-\frac{{\zeta }^{2}t}{2}}{\left(t\varphi +\frac{x^{\omega }}{\omega }\right)}^2}{\varphi {\left(-\frac{3{\beta }_{3}C{w}_2^2\varphi +3w_2^2{\varphi }^2-4{\beta }_1{\beta }_3{w}_3^2-8{\beta }_1{\beta }_4{w}_3^2+3{\beta }_2{w}_3^2\varphi }{3\varphi ({\beta }_{3}C+\varphi )}+w_3^2{\left(t\varphi +\frac{x^{\omega }}{\omega }\right)}^2+w_2^2\right)}^2},\end{array}$$

(38)

$$\begin{array}{c}\hfill v(x,t)=\frac{4w_3^2{e}^{\zeta \eta \left(t\right)-\frac{{\zeta }^{2}t}{2}}\left(t\varphi +\frac{x^{\omega }}{\omega }\right)}{-\frac{3{\beta }_{3}C{w}_2^2\varphi +3w_2^2{\varphi }^2-4{\beta }_1{\beta }_3{w}_3^2-8{\beta }_1{\beta }_4{w}_3^2+3{\beta }_2{w}_3^2\varphi }{3\varphi ({\beta }_{3}C+\varphi )}+w_3^2{\left(t\varphi +\frac{x^{\omega }}{\omega }\right)}^2+w_2^2}.\end{array}$$

(39)

8. M-Shaped Rational Wave Solution with One Kink Wave

For an M-shaped rational wave solution with one kink wave, we assume the following f [29]:

$$\begin{array}{cc}& f=c_1^2+c_2^2+r_1{e}^{G_1}+w_5,\\ \hfill & c_1=w_1\nu +w_2,c_2=w_3\nu +w_4,\\ \hfill & G_1=l_1\nu +l_2.\end{array}$$

(40)

Using Equation (40) into Equation (15) and setting the coefficients of $x,t,e^{3l_1\nu +3l_2},e^{2l_1\nu +2l_2}$ and $e^{l_1\nu +l_2}$ to zero, we have some values for the wave solution:

$$\begin{array}{c}\hfill w_1=0,l_1=\sqrt{-\frac{{\beta }_{3}C+\varphi }{{\beta }_2}},w_4=\frac{w_3\sqrt{-\frac{{\beta }_{3}C+\varphi }{{\beta }_2}}(4{\beta }_1{\beta }_3+8{\beta }_1{\beta }_4-9{\beta }_2\varphi )}{15\left(\varphi ({\beta }_{3}C+\varphi )\right)}.\end{array}$$

(41)

Evaluating Equation (41) into Equation (40) and then putting the result into Equation (14), we have

$$\begin{array}{c}\hfill V=\frac{2\left(r_1\sqrt{\frac{{\beta }_3(-C)-\varphi }{{\beta }_2}}{e}^{\nu \sqrt{\frac{{\beta }_3(-C)-\varphi }{{\beta }_2}}+l_2}+2w_3\left(\frac{w_3\sqrt{\frac{{\beta }_3(-C)-\varphi }{{\beta }_2}}(4{\beta }_1{\beta }_3+8{\beta }_1{\beta }_4-9{\beta }_2\varphi )}{15\varphi ({\beta }_{3}C+\varphi )}+\nu w_3\right)\right)}{r_1{e}^{\nu \sqrt{\frac{{\beta }_3(-C)-\varphi }{{\beta }_2}}+l_2}+{\left(\frac{w_3\sqrt{\frac{{\beta }_3(-C)-\varphi }{{\beta }_2}}(4{\beta }_1{\beta }_3+8{\beta }_1{\beta }_4-9{\beta }_2\varphi )}{15\varphi ({\beta }_{3}C+\varphi )}+\nu w_3\right)}^2+w_2^2+w_5}.\end{array}$$

(42)

Substituting Equation (42) into Equation (11), we get

$$\begin{array}{c}\hfill U=-\frac{4{\beta }_{1}C{\left(r_1\sqrt{\frac{{\beta }_3(-C)-\varphi }{{\beta }_2}}{e}^{\nu \sqrt{\frac{{\beta }_3(-C)-\varphi }{{\beta }_2}}+l_2}+2w_3\left(\frac{w_3\sqrt{\frac{{\beta }_3(-C)-\varphi }{{\beta }_2}}(4{\beta }_1{\beta }_3+8{\beta }_1{\beta }_4-9{\beta }_2\varphi )}{15\varphi ({\beta }_{3}C+\varphi )}+\nu w_3\right)\right)}^2}{\varphi {\left(r_1{e}^{\nu \sqrt{\frac{{\beta }_3(-C)-\varphi }{{\beta }_2}}+l_2}+{\left(\frac{w_3\sqrt{\frac{{\beta }_3(-C)-\varphi }{{\beta }_2}}(4{\beta }_1{\beta }_3+8{\beta }_1{\beta }_4-9{\beta }_2\varphi )}{15\varphi ({\beta }_{3}C+\varphi )}+\nu w_3\right)}^2+w_2^2+w_5\right)}^2}.\end{array}$$

(43)

Evaluating Equations (42) and (43) into Equation (3), we obtain the solutions given below

$$\begin{array}{c}\hfill u(x,t)=-\frac{4{\beta }_{1}C{e}^{\zeta \eta \left(t\right)-\frac{{\zeta }^{2}t}{2}}{\left(r_1\sqrt{\frac{{\beta }_3(-C)-\varphi }{{\beta }_2}}{e}^{\sqrt{\frac{{\beta }_3(-C)-\varphi }{{\beta }_2}}\left(t\varphi +\frac{x^{\omega }}{\omega }\right)+l_2}+2w_3\left(\mathsf{\Pi }\right)\right)}^2}{\varphi {\left(r_1{e}^{\sqrt{\frac{{\beta }_3(-C)-\varphi }{{\beta }_2}}\left(t\varphi +\frac{x^{\omega }}{\omega }\right)+l_2}+{\left(\mathsf{\Pi }\right)}^2+w_2^2+w_5\right)}^2},\end{array}$$

(44)

$$\begin{array}{c}\hfill v(x,t)=\frac{2e^{\zeta \eta \left(t\right)-\frac{{\zeta }^{2}t}{2}}\left(r_1\sqrt{\frac{{\beta }_3(-C)-\varphi }{{\beta }_2}}{e}^{\sqrt{\frac{{\beta }_3(-C)-\varphi }{{\beta }_2}}\left(t\varphi +\frac{x^{\omega }}{\omega }\right)+l_2}+2w_3\left(\mathsf{\Pi }\right)\right)}{r_1{e}^{\sqrt{\frac{{\beta }_3(-C)-\varphi }{{\beta }_2}}\left(t\varphi +\frac{x^{\omega }}{\omega }\right)+l_2}+{\left(\mathsf{\Pi }\right)}^2+w_2^2+w_5},\end{array}$$

(45)

where $\mathsf{\Pi }=\frac{w_3\sqrt{\frac{{\beta }_3(-C)-\varphi }{{\beta }_2}}(4{\beta }_1{\beta }_3+8{\beta }_1{\beta }_4-9{\beta }_2\varphi )}{15\varphi ({\beta }_{3}C+\varphi )}+w_3\left(t\varphi +\frac{x^{\omega }}{\omega }\right).$

9. M-Shaped Rational Wave Solution with Two Kink Waves

For the M-shaped rational wave solution with two kink waves, we assume the following ansatz [30]:

$$\begin{array}{cc}& f=c_1^2+c_2^2+r_1{e}^{G_1}+r_2{e}^{G_2}+w_5,\\ \hfill & c_1=w_1\nu +w_2,c_2=w_3\nu +w_4,\\ \hfill & G_1=l_1\nu +l_2,G_2=l_3\nu +l_4.\end{array}$$

(46)

Inserting Equation (46) into Equation (15) and setting the coefficients of $x,t,e^{3l_1\nu +3l_2},e^{2l_1\nu +2l_2},e^{l_1\nu +l_2},e^{3l_3\nu +3l_4},e^{2l_3\nu +2l_4},e^{l_3\nu +l_4},e^{2l_1\nu +2l_2+l_3\nu +l_4},e^{l_1\nu +l_2+2l_3\nu +2l_4},e^{l_1\nu +l_2+l_3\nu +l_4}$ to zero, we get some values for the wave solution:

$$\begin{array}{c}\hfill l_3=0,w_1=0,w_5=-\frac{{\beta }_{3}C{w}_2^2+3{\beta }_{3}C{w}_4^2+w_2^2\varphi -3{\beta }_2{w}_3^2+3w_4^2\varphi }{{\beta }_{3}C+\varphi }.\end{array}$$

(47)

Putting Equation (47) into Equation (46) and then inserting into Equation (14), we get

$$\begin{array}{c}\hfill V=\frac{2\left(l_1{r}_1{e}^{l_1\nu +l_2}+2w_3(\nu w_3+w_4)\right)}{\frac{{\beta }_3(-C)w_2^2-3{\beta }_{3}C{w}_4^2-w_2^2\varphi +3{\beta }_2{w}_3^2-3w_4^2\varphi }{{\beta }_{3}C+\varphi }+r_1{e}^{l_1\nu +l_2}+e^{l_4}{r}_2+w_2^2+{(\nu w_3+w_4)}^2}.\end{array}$$

(48)

Inserting Equation (48) into Equation (11) yields

$$\begin{array}{c}\hfill U=-\frac{4\beta 1C{\left(l_1{r}_1{e}^{l_1\nu +l_2}+2w_3(\nu w_3+w_4)\right)}^2}{\varphi {\left(\frac{{\beta }_3(-C)w_2^2-3{\beta }_{3}C{w}_4^2-w_2^2\varphi +3{\beta }_2{w}_3^2-3w_4^2\varphi }{{\beta }_{3}C+\varphi }+r_1{e}^{l_1\nu +l_2}+e^{l_4}{r}_2+w_2^2+{(\nu w_3+w_4)}^2\right)}^2}.\end{array}$$

(49)

Then, putting Equations (48) and (49) into Equation (3), we obtain

$$\begin{array}{c}\hfill u(x,t)=-\frac{4\beta 1C{e}^{\zeta \eta \left(t\right)-\frac{{\zeta }^{2}t}{2}}{\left(l_1{r}_1{e}^{l_1\left(t\varphi +\frac{x^{\omega }}{\omega }\right)+l_2}+2w_3\left(w_3\left(t\varphi +\frac{x^{\omega }}{\omega }\right)+w_4\right)\right)}^2}{\varphi {\left(\frac{{\beta }_3(-C)w_2^2-3{\beta }_{3}C{w}_4^2-w_2^2\varphi +3{\beta }_2{w}_3^2-3w_4^2\varphi }{{\beta }_{3}C+\varphi }+r_1{e}^{l_1\left(t\varphi +\frac{x^{\omega }}{\omega }\right)+l_2}+e^{l_4}{r}_2+{\left(w_3\left(t\varphi +\frac{x^{\omega }}{\omega }\right)+w_4\right)}^2+w_2^2\right)}^2},\end{array}$$

(50)

$$\begin{array}{c}\hfill v(x,t)=\frac{2e^{\zeta \eta \left(t\right)-\frac{{\zeta }^{2}t}{2}}\left(l_1{r}_1{e}^{l_1\left(t\varphi +\frac{x^{\omega }}{\omega }\right)+l_2}+2w_3\left(w_3\left(t\varphi +\frac{x^{\omega }}{\omega }\right)+w_4\right)\right)}{\frac{{\beta }_3(-C)w_2^2-3{\beta }_{3}C{w}_4^2-w_2^2\varphi +3{\beta }_2{w}_3^2-3w_4^2\varphi }{{\beta }_{3}C+\varphi }+r_1{e}^{l_1\left(t\varphi +\frac{x^{\omega }}{\omega }\right)+l_2}+e^{l_4}{r}_2+{\left(w_3\left(t\varphi +\frac{x^{\omega }}{\omega }\right)+w_4\right)}^2+w_2^2}.\end{array}$$

(51)

10. M-Shaped Interaction with Rogue and Kink Waves

For the M-shaped interaction with rogue and kink waves, we assume the following f [29]:

$$\begin{array}{cc}& f=c_1^2+c_2^2++r_{1}cosh\left(G_1\right)+r_2{e}^{G_2}+w_5,\\ \hfill & c_1=w_1\nu +w_2,c_2=w_3\nu +w_4,\\ \hfill & G_1=l_1\nu +l_2,G_2=l_3\nu +l_4.\end{array}$$

(52)

By using Equation (52) into Equation (15) and setting the coefficients of $x,t,e^{3l_3\nu +3l_4},e^{2l_3\nu +2l_4},$$e^{l_3\nu +l_4},cosh(l_1\nu +l_2),cosh(l_1\nu +l_2),e^{l_3\nu +l_4}cosh(l_1\nu +l_2),{cosh}^2(l_1\nu +l_2),sinh(l_1\nu +l_2),$$e^{2l_3\nu +2l_4}sinh(l_1\nu +l_2),e^{l_3\nu +l_4}sinh(l_1\nu +l_2),sinh(l_1\nu +l_2)cosh(l_1\nu +l_2),$$sinh(l_1\nu +l_2){cosh}^2(l_1\nu +l_2),e^{l_3\nu +l_4}{sinh}^2(l_1\nu +l_2),{sinh}^2(l_1\nu +l_2)$ and ${sinh}^3(l_1\nu +l_2)$ to zero, we are left with some values for the wave solution:

$$\begin{array}{c}\hfill w_1=0,l_3=\sqrt{-\frac{{\beta }_{3}C+\varphi }{{\beta }_2}},w_4=\frac{w_3(4{\beta }_1{\beta }_3-9{\beta }_2\varphi )\sqrt{-\frac{{\beta }_{3}C+\varphi }{{\beta }_2}}}{15\left(\varphi ({\beta }_{3}C+\varphi )\right)}.\end{array}$$

(53)

Putting Equation (53) into Equation (52) and then into Equation (14), we have the solution

$$\begin{array}{c}\hfill V=\frac{2\left(\xi e^{\nu \sqrt{\frac{{\beta }_3(-C)-\nu }{{\beta }_2}}+l_4}+2\left(\chi \right)\left(\Delta +w_3\right)+l_1{r}_{1}sinh(l_1\nu +l_2)\right)}{r_2{e}^{\nu \sqrt{\frac{{\beta }_3(-C)-\nu }{{\beta }_2}}+l_4}+{\left(\chi \right)}^2+r_{1}cosh(l_1\nu +l_2)+w_2^2+w_5},\end{array}$$

(54)

$$\begin{array}{c}\hfill U=-\frac{4{\beta }_{1}C{\left(\xi e^{\nu \sqrt{\frac{{\beta }_3(-C)-\nu }{{\beta }_2}}+l_4}+2\left(\chi \right)\left(\Delta +w_3\right)+l_1{r}_{1}sinh(l_1\nu +l_2)\right)}^2}{\varphi {\left(r_2{e}^{\nu \sqrt{\frac{{\beta }_3(-C)-\nu }{{\beta }_2}}+l_4}+{\left(\chi \right)}^2+r_{1}cosh(l_1\nu +l_2)+w_2^2+w_5\right)}^2},\end{array}$$

(55)

where $\xi =r_2\left(\sqrt{\frac{{\beta }_3(-C)-\nu }{{\beta }_2}}-\frac{\nu }{2{\beta }_2\sqrt{\frac{{\beta }_3(-C)-\nu }{{\beta }_2}}}\right)$, $\chi =\frac{w_3(4{\beta }_1{\beta }_3-9{\beta }_2\nu )\sqrt{\frac{{\beta }_3(-C)-\nu }{{\beta }_2}}}{15\nu ({\beta }_{3}C+\nu )}+\nu w_3$ and $\Delta =-\frac{w_3(4{\beta }_1{\beta }_3-9{\beta }_2\nu )\sqrt{\frac{{\beta }_3(-C)-\nu }{{\beta }_2}}}{15{\nu }^2({\beta }_{3}C+\nu )}-\frac{w_3(4{\beta }_1{\beta }_3-9{\beta }_2\nu )}{30{\beta }_2\nu ({\beta }_{3}C+\nu )\sqrt{\frac{{\beta }_3(-C)-\nu }{{\beta }_2}}}-\frac{w_3(4{\beta }_1{\beta }_3-9{\beta }_2\nu )\sqrt{\frac{{\beta }_3(-C)-\nu }{{\beta }_2}}}{15\nu {({\beta }_{3}C+\nu )}^2}-\frac{3{\beta }_2{w}_3\sqrt{\frac{{\beta }_3(-C)-\nu }{{\beta }_2}}}{5\nu ({\beta }_{3}C+\nu )}.$

Evaluating Equations (54) and (55) into Equation (3), we obtain

$$\begin{array}{c}\hfill u(x,t)=-\frac{4{\beta }_{1}C{e}^{\zeta \eta \left(t\right)-\frac{{\zeta }^{2}t}{2}}{\left(r_2\left(\sqrt{\frac{-{\beta }_{3}C-t\varphi -\frac{x^{\omega }}{\omega }}{{\beta }_2}}-\frac{t\varphi +\frac{x^{\omega }}{\omega }}{2{\beta }_2\sqrt{\frac{-{\beta }_{3}C-t\varphi -\frac{x^{\omega }}{\omega }}{{\beta }_2}}}\right)e^{\lambda }+2\left(\mathsf{\Gamma }-\mathsf{\Lambda }\right)\left(\varrho \right)+l_1{r}_{1}sinh\left(\theta \right)\right)}^2}{\varphi {\left(r_2{e}^{\lambda }+{\left(\varrho \right)}^2+r_{1}cosh\left(\theta \right)+w_2^2+w_5\right)}^2},\end{array}$$

(56)

$$\begin{array}{c}\hfill v(x,t)=\frac{2e^{\zeta \eta \left(t\right)-\frac{{\zeta }^{2}t}{2}}\left(r_2\left(\sqrt{\frac{-{\beta }_{3}C-t\varphi -\frac{x^{\omega }}{\omega }}{{\beta }_2}}-\frac{t\varphi +\frac{x^{\omega }}{\omega }}{2{\beta }_2\sqrt{\frac{-{\beta }_{3}C-t\varphi -\frac{x^{\omega }}{\omega }}{{\beta }_2}}}\right)e^{\left(\lambda \right)}+2\left(\mathsf{\Gamma }+\mathsf{\Lambda }\right)\left(\varrho \right)+l_1{r}_{1}sinh\left(\theta \right)\right)}{r_2{e}^{\left(\lambda \right)}+{\left(\varrho \right)}^2+r_{1}cosh\left(\theta \right)+w_2^2+w_5},\end{array}$$

(57)

where $\varrho =\frac{w_3\sqrt{\frac{-{\beta }_{3}C-t\varphi -\frac{x^{\omega }}{\omega }}{{\beta }_2}}\left(4{\beta }_1{\beta }_3-9{\beta }_2\left(t\varphi +\frac{x^{\omega }}{\omega }\right)\right)}{15\left(t\varphi +\frac{x^{\omega }}{\omega }\right)\left({\beta }_{3}C+t\varphi +\frac{x^{\omega }}{\omega }\right)}+w_3\left(t\varphi +\frac{x^{\omega }}{\omega }\right)$, $\theta =l_1\left(t\varphi +\frac{x^{\omega }}{\omega }\right)+l_2$,

$\mathsf{\Gamma }=-\frac{w_3\left(4{\beta }_1{\beta }_3-9{\beta }_2\left(t\varphi +\frac{x^{\omega }}{\omega }\right)\right)}{30{\beta }_2\left(t\varphi +\frac{x^{\omega }}{\omega }\right)\left({\beta }_{3}C+t\varphi +\frac{x^{\omega }}{\omega }\right)\sqrt{\frac{-{\beta }_{3}C-t\varphi -\frac{x^{\omega }}{\omega }}{{\beta }_2}}}-\frac{w_3\sqrt{\frac{-{\beta }_{3}C-t\varphi -\frac{x^{\omega }}{\omega }}{{\beta }_2}}\left(4{\beta }_1{\beta }_3-9{\beta }_2\left(t\varphi +\frac{x^{\omega }}{\omega }\right)\right)}{15{\left(t\varphi +\frac{x^{\omega }}{\omega }\right)}^2\left({\beta }_{3}C+t\varphi +\frac{x^{\omega }}{\omega }\right)}$,

$\mathsf{\Lambda }=-\frac{w_3\sqrt{\frac{-{\beta }_{3}C-t\varphi -\frac{x^{\omega }}{\omega }}{{\beta }_2}}\left(4{\beta }_1{\beta }_3-9{\beta }_2\left(t\varphi +\frac{x^{\omega }}{\omega }\right)\right)}{15\left(t\varphi +\frac{x^{\omega }}{\omega }\right){\left({\beta }_{3}C+t\varphi +\frac{x^{\omega }}{\omega }\right)}^2}-\frac{3{\beta }_2{w}_3\sqrt{\frac{-{\beta }_{3}C-t\varphi -\frac{x^{\omega }}{\omega }}{{\beta }_2}}}{5\left(t\varphi +\frac{x^{\omega }}{\omega }\right)\left({\beta }_{3}C+t\varphi +\frac{x^{\omega }}{\omega }\right)}+w_3$,

$\lambda =\left(t\varphi +\frac{x^{\omega }}{\omega }\right)\sqrt{\frac{-{\beta }_{3}C-t\varphi -\frac{x^{\omega }}{\omega }}{{\beta }_2}}+l_4$.

11. M-Shaped Interaction with Periodic and Kink Waves

For the M-shaped interaction with periodic and kink waves, we use the given transformation [31,32,33]:

$$\begin{array}{cc}& f=c_1^2+c_2^2++r_{1}cos\left(G_1\right)+r_2{e}^{G_2}+w_5,\\ \hfill & c_1=w_1\nu +w_2,c_2=w_3\nu +w_4,\\ \hfill & G_1=l_1\nu +l_2,G_2=l_3\nu +l_4.\end{array}$$

(58)

By using Equation (58) into Equation (15) and by comparing the coefficients of $x,t,e^{3l_3\nu +3l_4},e^{2l_3\nu +2l_4},e^{l_3\nu +l_4},cos(l_1\nu +l_2),e^{l_3\nu +l_4}cos(l_1\nu +l_2),{cos}^2(l_1\nu +l_2),e^{l_3\nu +l_4}{cos}^2(l_1\nu +l_2),$$e^{2l_3\nu +2l_4}sin(l_1\nu +l_2),sin(l_1\nu +l_2),e^{l_3\nu +l_4}sin(l_1\nu +l_2),e^{l_3\nu +l_4}sin(l_1\nu +l_2)cos(l_1\nu +l_2),$$sin(l_1\nu +l_2)cos(l_1\nu +l_2),sin(l_1\nu +l_2){cos}^2(l_1\nu +l_2)$ and ${sin}^3(l_1\nu +l_2)$, we get some values for the wave solution:

$$\begin{array}{c}\hfill l_3=0,w_1=-\frac{w_3{w}_4}{w_2},\\ \hfill w_5=-\frac{{\beta }_{3}C{w}_2^4+{\beta }_{3}C{w}_2^2{w}_4^2+w_2^4\varphi -3{\beta }_2{w}_2^2{w}_3^2+w_2^2{w}_4^2\varphi -3{\beta }_2{w}_3^2{w}_4^2}{w_2^2({\beta }_{3}C+\varphi )}.\end{array}$$

(59)

After inserting Equation (59) into Equation (58) and then inserting into Equation (14), we get

$$\begin{array}{c}\hfill V=\frac{2\left(-\frac{\gamma }{w_2^2{({\beta }_{3}C+\nu )}^2}+\frac{-w_2^4-w_2^2{w}_4^2}{w_2^2({\beta }_{3}C+\nu )}-l_1{r}_{1}sin(l_1\nu +l_2)-\frac{2w_3{w}_4\left(w_2-\frac{\nu w_3{w}_4}{w_2}\right)}{w_2}+2w_3(\nu w_3+w_4)\right)}{\frac{\gamma }{w_2^2({\beta }_{3}C+\nu )}+r_{1}cos(l_1\nu +l_2)+e^{l_4}{r}_2+{\left(w_2-\frac{\nu w_3{w}_4}{w_2}\right)}^2+{(\nu w_3+w_4)}^2},\end{array}$$

(60)

$$\begin{array}{c}\hfill U=-\frac{4{\beta }_{1}C{\left(-\frac{\gamma }{w_2^2{({\beta }_{3}C+\nu )}^2}+\frac{-w_2^4-w_2^2{w}_4^2}{w_2^2({\beta }_{3}C+\nu )}-l_1{r}_{1}sin(l_1\nu +l_2)-\frac{2w_3{w}_4\left(w_2-\frac{\nu w_3{w}_4}{w_2}\right)}{w_2}+2w_3(\nu w_3+w_4)\right)}^2}{\varphi {\left(\frac{\gamma }{w_2^2({\beta }_{3}C+\nu )}+r_{1}cos(l_1\nu +l_2)+e^{l_4}{r}_2+{\left(w_2-\frac{\nu w_3{w}_4}{w_2}\right)}^2+{(\nu w_3+w_4)}^2\right)}^2},\end{array}$$

(61)

where $\gamma ={\beta }_3(-C)w_2^4-{\beta }_{3}C{w}_2^2{w}_4^2-\nu w_2^4+3{\beta }_2{w}_2^2{w}_3^2-\nu w_2^2{w}_4^2+3{\beta }_2{w}_3^2{w}_4^2.$

By putting Equations (60) and (61), we have the following solutions

$$\begin{array}{c}\hfill u(x,t)=-\frac{4\beta 1C{e}^{\zeta \eta \left(t\right)-\frac{{\zeta }^{2}t}{2}}{\left(-\frac{\varsigma }{w_2^2{\left({\beta }_{3}C+t\varphi +\frac{x^{\omega }}{\omega }\right)}^2}+\frac{-w_2^4-w_2^2{w}_4^2}{w_2^2\left({\beta }_{3}C+t\varphi +\frac{x^{\omega }}{\omega }\right)}-l_1{r}_{1}sin\left(\varpi \right)-\frac{2w_3{w}_4\left(\psi \right)}{w_2}+2w_3\left(\kappa \right)\right)}^2}{\varphi {\left(\frac{\varsigma }{w_2^2\left({\beta }_{3}C+t\varphi +\frac{x^{\omega }}{\omega }\right)}+r_{1}cos\left(\varpi \right)+e^{l_4}{r}_2+{\left(\psi \right)}^2+{\left(\kappa \right)}^2\right)}^2},\end{array}$$

(62)

$$\begin{array}{c}\hfill v(x,t)=\frac{2e^{\zeta \eta \left(t\right)-\frac{{\zeta }^{2}t}{2}}\left(-\frac{\varsigma }{w_2^2{\left({\beta }_{3}C+t\varphi +\frac{x^{\omega }}{\omega }\right)}^2}+\frac{-w_2^4-w_2^2{w}_4^2}{w_2^2\left({\beta }_{3}C+t\varphi +\frac{x^{\omega }}{\omega }\right)}-l_1{r}_{1}sin\left(\varpi \right)-\frac{2w_3{w}_4\left(\psi \right)}{w_2}+2w_3\left(\kappa \right)\right)}{\frac{\varsigma }{w_2^2\left({\beta }_{3}C+t\varphi +\frac{x^{\omega }}{\omega }\right)}+r_{1}cos\left(\varpi \right)+e^{l_4}{r}_2+{\left(\psi \right)}^2+{\left(\kappa \right)}^2},\end{array}$$

(63)

where $\varsigma ={\beta }_3(-C)w_2^4-{\beta }_{3}C{w}_2^2{w}_4^2-w_2^4\left(t\varphi +\frac{x^{\omega }}{\omega }\right)-w_2^2{w}_4^2\left(t\varphi +\frac{x^{\omega }}{\omega }\right)+3{\beta }_2{w}_2^2{w}_3^2+3{\beta }_2{w}_3^2{w}_4^2$, $\varpi =l_1\left(t\varphi +\frac{x^{\omega }}{\omega }\right)+l_2$, $\psi =w_2-\frac{w_3{w}_4\left(t\varphi +\frac{x^{\omega }}{\omega }\right)}{w_2}$ and $\kappa =w_3\left(t\varphi +\frac{x^{\omega }}{\omega }\right)+w_4$.

12. Results and Discussion

Some researchers worked on the governing model such as Askar et al., who used the (G’/G)-expansion method to find exact solutions for the fractional–stochastic Drinfel’d–Sokolov–Wilson equations [18]. Qin and Yan worked on the applications of the coupled Drinfel’d–Sokolov–Wilson equation and also used an improved F-expansion method to find exact doubly periodic solutions in terms of the rational formal Jacobi elliptic function of nonlinear partial differential equations [34].

By selecting the appropriate values for the parameter, we were able to generate the desired types of solution that indicated a wave discrepancy. In Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27 and Figure 28, we presented 3D, 2D, contour plots, respectively. In Figure 1, the M and W shape waves appeared with bright and dark faces. In Figure 2, we obtained a bright face and after some time, bright–dark faces appeared; in Figure 3 and Figure 4, we represented 2D and contour plots of this wave solution by using the values ${\beta }_2=0.2,{\beta }_3=3.5,\zeta =0,l_2=0.03,Q=5,r_1=2.5,w_2=5,$$w_3=0.3,w_4=0.9,\omega =0.1$ and $\varphi =1.7$. According to Equations (26) and (27), the periodic waves produced in Figure 5, Figure 6, Figure 7 and Figure 8 varied in amplitude. In Figure 9, Figure 10 and Figure 11, we can see one stripe soliton propagating at different times. The MSR solution was shown in Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16, where M-shaped waves appeared with bright–dark faces. In Figure 17a and Figure 18a, one kink wave appeared and after some time, that one kink wave changed into the M-shaped wave shown in Figure 17b and Figure 18b for Equation (44) and the remaining figures for that solution showed the 2D and contour plots. The MSR solution with two kink waves in Figure 19, Figure 20 and Figure 21 with bright and dark faces was derived from Equations (50) and (51). For Equations (56) and (57) and Equations (62) and (63), the M-shaped interactions with RK and PK with some M-shaped dark and bright faces are shown in Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27 and Figure 28.

Figure 1. Show the behaviour of $u(x,t)$ in Equation (20), it is presented with ${\beta }_1=3.5,{\beta }_2=0.2,$${\beta }_3=1.5,$$\zeta =0,l_2=0.3,Q=2,r_1=1.5,w_2=2.5,w_3=7.3,w_4=1.9$ and $\varphi =4.7.$ (a,b) shows 3D graphs presenting bright and dark faces for $\omega =0.8$ and $\omega =0.5$ respectively, (c,d) show 2D graphs for bright and dark faces for $\omega =0.8$ and $\omega =0.5$ respectively. (e,f) show contour graphs for bright and dark faces for $\omega =0.8$ and $\omega =0.5$ respectively.

Figure 2. (a–c) show three-dimensional plots.

Figure 3. (a–c) show two-dimensional plots.

Figure 4. Graphical demonstration of the two-dimensional and three-dimensional representations and contour of solution $v(x,t)$ in Equation (21) with ${\beta }_2=0.2,{\beta }_3=3.5,\zeta =0,l_2=0.03,Q=5,r_1=2.5,w_2=5,w_3=0.3,w_4=0.9,$ and $\varphi =1.7.$ (a–c) show contour graphs.

Figure 5. (a,b) show 3D plots and (c,d) show two-dimensional plots.

Figure 6. (a,b) show contour plots. The graphical behaviour of $u(x,t)$ in Equation (26) is presented with ${\beta }_1=0.3,{\beta }_2=0.2,{\beta }_3=0.5,{\beta }_4=0.4,\zeta =0,l_1=0.3,l_2=0.1,l_4=4.5,l_5=-0.5,$$Q=1.2,r_1=2.5,r_2=3.2,w_2=0.2,w_3=3.1,w_4=1.5,w_5=5$ and $\varphi =1.4.$

Figure 7. (a,b) show three-dimensional plots and (c,d) show two-dimensional plots.

Figure 8. Graphical demonstration of solution $v(x,t)$ in Equation (27) presented with ${\beta }_1=0.3,{\beta }_2=0.2,$${\beta }_3=0.5,{\beta }_4=0.04,\zeta =0,l_1=0.1,l_2=0.01,l_4=4.4,l_5=0.5,Q=1.2,$$r_1=2.5,r_2=3.2,w_2=0.02,w_3=3.1,w_4=1.5,w_5=5$ and $\varphi =1.4.$ (a,b) show contour plots for various values of $\omega .$

Figure 9. (a,b) show 3D plots and (c,d) show 2D plots.

Figure 10. Dynamical behaviour of $u(x,t)$ in Equation (32) presented with ${\beta }_1=1.2,$${\beta }_2=0.2,{\beta }_3=3.5,\zeta =0,l_2=0.3,l_4=4.2,l_6=1.6,Q=0.5,r_1=1.5,r_2=2.5,w=3,w_1=2.5,$ and $\varphi =1.3.$ (a,b) show contour plots for various values of $\omega .$

Figure 11. (a–c) show 3D graphs. Dynamical representation of solution $v(x,t)$ in Equation (33) with ${\beta }_2=0.2,{\beta }_3=3.5,$$\zeta =0,l_2=0.03,Q=5,r_1=2.5,w_2=5,w_3=0.3,w_4=0.9,\varphi =1.7.$ (d–f) show 2D plots and (g–i) represent contour graphs for various values of $\omega$.

Figure 12. (a,b) show 3D plots.

Figure 13. Dynamical behaviour of $u(x,t)$ in Equation (38) presented with ${\beta }_1=0.3,{\beta }_2=2.2,$${\beta }_3=0.5,{\beta }_4=5.4,\zeta =0,Q=2,w_2=1.1,w_3=3.2$ and $\varphi =1.4.$ (a,b) show 2D plots and (c,d) represent contour plots for different values of $\omega$.

Figure 14. (a–c) show 3D plots.

Figure 15. (a–c) show 2D plots.

Figure 16. Behaviour of $v(x,t)$ in Equation (39) presented with ${\beta }_1=1.3,{\beta }_2=0.2,$${\beta }_3=1.5,{\beta }_4=0.4,\zeta =0,Q=5.2,w_2=0.1,w_3=3$ and $\varphi =1.4.$ (a–c) show contour graphs for various values of $\omega$.

Figure 17. Dynamical demonstration of $u(x,t)$ in Equation (44) presented with ${\beta }_1=1.3,{\beta }_2=0.2,$${\beta }_3=3.5,{\beta }_4=0.4,\zeta =0,l_2=0.1,l_5=0.5,Q=3.2,r_1=2.5,w_2=2,w_3=3.5,w_5=1.5,$ and $\varphi =5.4.$ (a,b) show 3D graph of lump wave, (c,d) represent 2D graph and (e,f) show contour plot.

Figure 18. Dynamical behaviour of $v(x,t)$ in Equation (45) represented via ${\beta }_1=0.3,$${\beta }_2=0.2,{\beta }_3=0.5,{\beta }_4=0.04,\zeta =0,l_2=0.01,l_5=0.5,Q=1.2,r_1=2.5,w_2=0.02,w_3=3.1,$ and $\varphi =1.4.$ (a,b) show 3D graph of lump wave, (c,d) represent 2D graph and (e,f) show contour plot for different values of $\omega$.

Figure 19. (a,b) show 3D plots and (c,d) show 2D plots.

Figure 20. Dynamical demonstration of $u(x,t)$ in Equation (50) presented with ${\beta }_1=1.5,{\beta }_2=0.2,$${\beta }_3=0.5,\zeta =0,l_1=1.1,l_2=0.1,l_4=0.5,Q=1.2,r_1=2.5,r_2=0.12,w_2=0.2,w_3=0.03,$$w_4=3.1$ and $\varphi =1.4.$ (a,b) show contour plots for different $\omega$.

Figure 21. Graphical representation of solution $v(x,t)$ in Equation (51) with ${\beta }_2=0.2,{\beta }_3=0.5,\zeta =0,$$l_1=1.1,l_2=0.01,l_4=0.5,Q=1.2,r_1=2.5,r_2=0.12,w_2=0.02,w_3=0.03,w_4=3.1,$ and $\varphi =1.4.$ (a,b) show 3D graph of lump wave, (c,d) represent 2D graph and (e,f) show contour plot.

Figure 22. Graphical demonstration of $u(x,t)$ in Equation (56) presented with ${\beta }_1=0.3,{\beta }_2=1.2,$${\beta }_3=0.5,\zeta =0,l_1=-1.8,l_2=0.05,l_4=0.01,Q=5.2,r_1=2.5,r_2=1.3,w_2=0.1,w_3=3,$$w_4=1.9,w_5=1.2$ and $\varphi =1.4.$ (a,b) show 3D graph, (c,d) represent 2D graph and (e,f) show contour plot.

Figure 23. (a,b) show 3D graphs and (c,d) represent 2D graph. Dynamical presentation of solution $v(x,t)$ in Equation (57) with ${\beta }_1=0.3,{\beta }_2=1.2,{\beta }_3=0.5,$$\zeta =0,l_1=-1.8,l_2=0.05,l_4=0.01,Q=5.2,r_1=2.5,r_2=1.3,w_2=0.1,w_3=3,w_4=1.9,$$w_5=1.2$ and $\varphi =1.4.$ (e,f) show contour plot.

Figure 24. (a,b) show 3D plots.

Figure 25. Graphical presentation of $u(x,t)$ in Equation (62) presented with ${\beta }_1=0.5,{\beta }_2=1.2,$${\beta }_3=2.5,\zeta =0,l_1=3,l_2=0.1,l_4=2,Q=5,r_1=1.5,r_2=3.1,w_2=1.1,w_3=2.3,w_4=0.9$ and $\varphi =1.2.$ (a,b) show 2D plot and (c,d) show contour graph.

Figure 26. (a,b) show 3D plots.

Figure 27. (a,b) show 2D plots.

Figure 28. Dynamical presentation of solution $v(x,t)$ in Equation (63) with ${\beta }_2=0.2,{\beta }_3=3.5,$$\zeta =0,l_1=0.03,l_2=0.01,l_4=2,Q=5,r_1=4.5,r_2=2.1,w_2=0.1,w_3=4.3,w_4=1.9$ and $\varphi =1.2.$ (a,b) show contour plot.

13. Conclusions

In this paper, we explored some wave solutions for stochastic–fractional Drinfel’d–Sokolov–Wilson. These equations are used in applied sciences, plasma physics, population dynamics, surface physics and mathematical physics. The obtained solutions were better and more useful and efficient for understanding a variety of significant physical phenomena. We acquired different types of solutions such as the periodic cross-rational wave solution, cross-kink rational wave solution, homoclinic breather wave solution, M-shaped rational wave solution, M-shaped rational wave solution with one kink wave, M-shaped rational wave solution with two kink waves, M-shaped interaction with rogue and kink waves, M-shaped interaction with periodic and kink waves. We also represented these wave solutions graphically.