Modulation Instability and Abundant Exact Solitons to the Fractional Mathematical Physics Model Through Two Distinct Methods

Abstract

The paper consists of various types of wave solutions for the truncated M-fractional Bateman–Burgers equation, a significant mathematical physics equation. This model describes the nonlinear waves and solitons in different physical fields such as optical fibers, plasma physics, fluid dynamics, traffic flow, etc. Through the application of the ${exp}_a$ function method and the modified simplest equation method, we are able to obtain exact series of soliton solutions. The results differ from the current solutions of the Bateman–Burgers model because of the fractional derivative. The achieved results could be helpful in various engineering and scientific domains. The Mathematica software is used to assist in obtaining and verifying the exact solutions and to obtain contour plots of the solutions in two and three dimensions. To ensure that the model in question is stable, a stability analysis is also carried out using the modulation instability method. Future research on the system in question and related systems will benefit from the findings. The methods used are simple and effective.

1. Introduction

Fractional calculus has gained much importance in different areas of science and engineering. Many phenomena are represented in fractional nonlinear partial differential equations (FNLPDEs). For example, the fractional Bogoyavlensky–Konopelchenko equation [1], the fractional Drinfeld–Sokolov equation [2], the fractional Kuralay equation [3], the fractional Zoomeron equation [4], the fractional Kadomtsev–Petviashvili equation [5], the fractional Fokas equation [6], the fractional higher-order Sasa–Satsuma equation [7], and many more. Many methods have been developed to solve the fractional nonlinear partial differential equations, for example, Lie symmetry analysis [8], improved modified extended tanh function method [9], improved generalized Riccati equation mapping method [10], new modified Sardar sub-equation method [11], improved generalized tanh-coth function method [12], and many more.

In our research work, we utilize the two useful and reliable methods: the ${exp}_a$ function method and the modified simplest equation (MSE) method. Each of the techniques has different uses, such as the Sasa–Satsuma equation [13], the Mikhailov–Novikov–Wang integrable equation [14], the Gross–Pitaevskii equation [15], etc. Similarly, the modified simplest equation method is utilized for the Sharma–Tasso–Olver equation [16], the Benjamin–Bona–Mahony equation [17], the Radha–Lakshmanan equation [18], the Whitham–Broer–Kaup equation [19].

Our concerned model in this research is the (3 + 1)-dimensional Bateman–Burgers model given as [20]

$$w_t-w{w}_x-w_{xx}-w_{yy}-w_{zz}=0.$$

(1)

Here, w is a wave function of spatio variables x, y, and z, and temporal variable t. This model describes the various physical phenomena, such as fluid dynamics, traffic flow, and nonlinear waves. It combines convection and diffusion. The competition between these two effects leads to the formation of shock waves, solitons, and other interesting phenomena. Different methods have been used for Equation (1), including the Sinc collection method [20], the bilinear neural network method [21], the simple equation method [22], the separation of variables [23], the developed Exp-function method [24], the Lie group analysis [25], the homotopy perturbation method [26], etc.

Our motivation was to study new types of truncated M-fractional exact solitons for the nonlinear (3 + 1)-dimensional Bateman–Burgers model by using two effective methods. Both the ${exp}_a$ function method and the modified simplest equation method provide different types of exact soliton solutions. These methods have not been previously applied for this model. The truncated M-fractional derivative (TMFD) gives solutions closer to the numerical solutions. This fractional derivative satisfies the conditions of both fractional derivatives and integer-order derivatives. The impact of the TMFD on the achieved results is displayed. The truncated M-fractional generalized water wave equation is the more prominent form of the model and helps us to understand the model more clearly. Soliton is defined as follows: “A soliton is a self-reinforcing solitary wave that maintains its shape and speed over long distances, despite interactions with other waves or perturbations. Solitons are stable solutions to certain nonlinear partial differential equations, exhibiting remarkable resilience against collisions and interactions.”

The paper consists of distinct sections given as follows: in Section 2, we explain the ${exp}_a$ function method and the modified simplest equation method; in Section 3, we give the mathematical analysis and utilize both methods to obtain exact solitons of the governing equation; in Section 4, we give the graphical description of some of our gained solutions; in Section 5, we provide modulation instability; and in Section 6, we give the conclusion about our complete research work.

Fractional Derivative

Definition 1.

Consider $w\left(y\right):[0,\infty )\to \Re$, so truncated M-fractional derivative (TMFD) of w of order ϵ is given as [27]

$$\begin{array}{c}\hfill D_{M,y}^{ϵ,\varrho }w\left(y\right)=\underset{ϵ\to 0}{lim}{\displaystyle \frac{w\left(y{E}_{\varrho }\left(ϵy^{1-ϵ}\right)\right)-w\left(y\right)}{ϵ}},ϵ\in (0,1],\varrho >0,\end{array}$$

and here, $E_{\varrho }(.)$ shows a truncated Mittag–Leffler (TML) profile given as [28]

$$\begin{array}{c}\hfill E_{\varrho }\left(z\right)=\sum _{j=0}^i{\displaystyle \frac{z^j}{\mathsf{\Gamma }(1+\varrho j)}},\varrho ispositiveandz\in \mathit{C}.\end{array}$$

Theorem 1.

Suppose a,b are real numbers, and g, f are both differentiable ϵ times for $y>0$, from [27]

$$\begin{array}{c}\hfill \left(a\right)D_{M,y}^{ϵ,\varrho }(ag\left(y\right)+bf\left(y\right))=a{D}_{M,y}^{ϵ,\varrho }g\left(y\right)+b{D}_{M,y}^{ϵ,\varrho }f\left(y\right).\end{array}$$

$$\begin{array}{c}\hfill \left(b\right)D_{M,y}^{ϵ,\varrho }(g\left(y\right).f\left(y\right))=g\left(y\right)D_{M,y}^{ϵ,\varrho }f\left(y\right)+f\left(y\right)D_{M,y}^{ϵ,\varrho }g\left(y\right).\end{array}$$

$$\begin{array}{c}\hfill \left(c\right)D_{M,y}^{ϵ,\varrho }\left({\displaystyle \frac{g\left(y\right)}{f\left(y\right)}}\right)={\displaystyle \frac{f\left(y\right)D_{M,y}^{ϵ,\varrho }g\left(y\right)-g\left(y\right)D_{M,y}^{ϵ,\varrho }f\left(y\right)}{{\left(f\left(y\right)\right)}^2}}.\end{array}$$

$$\begin{array}{c}\hfill \left(d\right)D_{M,y}^{ϵ,\varrho }\left(C\right)=0,\mathit{where}C\mathit{is}\mathrm{a}\mathit{constant}.\end{array}$$

$$\begin{array}{c}\hfill \left(e\right)D_{M,y}^{ϵ,\varrho }g\left(y\right)={\displaystyle \frac{y^{1-ϵ}}{\mathsf{\Gamma }(\varrho +1)}}{\displaystyle \frac{dg\left(y\right)}{dy}}.\end{array}$$

2. Methodologies

2.1. The ${exp}_a$ Function Method

Here we mention the basic steps of this method.

Assuming a nonlinear partial differential equation (NLPDE),

$$W(w,w^2,w_t,w_x,w_{tt},\dots )=0.$$

(2)

Equation (2) changes into nonlinear ordinary differential equation (NLODE)

$$Y(W,W^{\prime },W^{″},\dots )=0.$$

(3)

Consider the following wave relation:

$$w(x,t)=W\left(\mathsf{\Lambda }\right),\mathsf{\Lambda }=ax+ct.$$

(4)

Suppose the solution of Equation (3) is provided in [29,30,31]:

$$W\left(\mathsf{\Lambda }\right)={\displaystyle \frac{{\alpha }_0+{\alpha }_1{d}^{\mathsf{\Lambda }}+\dots +{\alpha }_m{d}^{m\mathsf{\Lambda }}}{{\beta }_0+{\beta }_1{d}^{\mathsf{\Lambda }}+\dots +{\beta }_m{d}^{m\mathsf{\Lambda }}}},d\ne 0,1.$$

(5)

Here, ${\alpha }_j$ and ${\beta }_j(0,1,2,3,\dots ,m)$ are unknown. By analyzing the homogeneous balance method in Equation (5), we gain m. Putting Equation (5) in Equation (3) provides

$$\wp \left(d^{\mathsf{\Lambda }}\right)=c_0+c_1{d}^{\mathsf{\Lambda }}+\dots +c_t{d}^{t\mathsf{\Lambda }}=0.$$

(6)

Putting $c_j(0,1,2,\dots ,t)$ in Equation (6) equal to zero, a set of equations is obtained:

$$c_j=0,herej=0,\dots ,t.$$

(7)

By solving the obtained set of equations, we can get the solutions of Equation (2).

There are some limitations to this method, including this method assumes a specific form for the solution, which might not always be valid. This method might not work for all types of nonlinear partial differential equations or might not provide a solution in some cases. This method relies on a specific exponential function form, which might not capture complex solution behaviors.

2.2. Description of Modified Simplest Equation (MSE) Method

Here we point out the some main phases of this method.

Phase 1:

Considering a NLPDE,

$$G(g,g^2{g}_x,g_t,g_{xx},g_{tt},g_{xt},\dots )=0,$$

(8)

where $g=g(x,t)$ shows a function.

Assuming the wave relation is given as

$$g(x,t)=V\left(\mathsf{\Lambda }\right),\mathsf{\Lambda }=x+\theta t.$$

(9)

Inserting Equation (9) in Equation (8), we obtain

$$Z(V,V^2{V}^{\prime },V^{″},\dots )=0.$$

(10)

Phase 2: Consider the solution of Equation (10),

$$V\left(\mathsf{\Lambda }\right)=b_0+\sum _{j=1}^m{b}_j{\psi }^j\left(\mathsf{\Lambda }\right),$$

(11)

where $b_j(j=1,2,\dots ,m)$ are undetermined.

Function $\varphi \left(\mathsf{\Lambda }\right)$ fulfills the given equation:

$${\varphi }^{\prime }\left(\mathsf{\Lambda }\right)={\varphi }^2\left(\mathsf{\Lambda }\right)+\eta ,$$

(12)

here $\eta$ is a constant.

Solutions of Equation (12) depend on $\eta$:

Type 1: If $\eta <0$,

$$\varphi \left(\mathsf{\Lambda }\right)=-\sqrt{-\eta }tanh\left(\sqrt{-\eta }\mathsf{\Lambda }\right),$$

(13)

$$\varphi \left(\mathsf{\Lambda }\right)=-\sqrt{-\eta }coth\left(\sqrt{-\eta }\mathsf{\Lambda }\right),$$

(14)

$$\varphi \left(\mathsf{\Lambda }\right)=\sqrt{-\eta }\left(-tanh\left(2\sqrt{-\eta }\mathsf{\Lambda }\right)\pm isech\left(2\sqrt{-\eta }\mathsf{\Lambda }\right)\right),$$

(15)

$$\varphi \left(\mathsf{\Lambda }\right)=\sqrt{-\eta }\left(-coth\left(2\sqrt{-\eta }\mathsf{\Lambda }\right)\pm csch\left(2\sqrt{-\eta }\mathsf{\Lambda }\right)\right),$$

(16)

$$\varphi \left(\mathsf{\Lambda }\right)=-{\displaystyle \frac{\sqrt{-\eta }}{2}}\left(tanh\left({\displaystyle \frac{\sqrt{-\eta }}{2}}\mathsf{\Lambda }\right)+coth\left({\displaystyle \frac{\sqrt{-\eta }}{2}}\mathsf{\Lambda }\right)\right).$$

(17)

Type 2: If $\eta >0$,

$$\varphi \left(\mathsf{\Lambda }\right)=\sqrt{\eta }tan\left(\sqrt{\eta }\mathsf{\Lambda }\right),$$

(18)

$$\varphi \left(\mathsf{\Lambda }\right)=-\sqrt{\eta }cot\left(\sqrt{\eta }\mathsf{\Lambda }\right),$$

(19)

$$\varphi \left(\mathsf{\Lambda }\right)=\sqrt{\eta }\left(tan\left(2\sqrt{\eta }\mathsf{\Lambda }\right)\pm sec\left(2\sqrt{\eta }\mathsf{\Lambda }\right)\right),$$

(20)

$$\varphi \left(\mathsf{\Lambda }\right)=\sqrt{\eta }\left(-cot\left(2\sqrt{\eta }\mathsf{\Lambda }\right)\pm csc\left(2\sqrt{\eta }\mathsf{\Lambda }\right)\right),$$

(21)

$$\varphi \left(\mathsf{\Lambda }\right)={\displaystyle \frac{\sqrt{\eta }}{2}}\left(tan\left({\displaystyle \frac{\sqrt{\eta }}{2}}\mathsf{\Lambda }\right)-cot\left({\displaystyle \frac{\sqrt{\eta }}{2}}\mathsf{\Lambda }\right)\right).$$

(22)

Type 3: If $\eta =0$,

$$\varphi \left(\mathsf{\Lambda }\right)=-{\displaystyle \frac{1}{\mathsf{\Lambda }}}.$$

(23)

There are some limitations for this method, for example, the choice of auxiliary equation can significantly impact the solution, and finding the right auxiliary equation can be difficult. Solutions obtained by using this method might not be general solutions but rather specific solutions. This method can become cumbersome for high-dimensional problems.

3. Mathematical Analysis and Exact Wave Solutions

$$D_{M,t}^{ϵ,\varrho }w-w{D}_{M,x}^{ϵ,\varrho }w-D_{M,x}^{2ϵ,\varrho }w-D_{M,y}^{2ϵ,\varrho }w-D_{M,z}^{2ϵ,\varrho }w=0.$$

(24)

$$w(x,t)=W\left(\mathsf{\Lambda }\right),\mathsf{\Lambda }={\displaystyle \frac{\mathsf{\Gamma }(1+\varrho )}{ϵ}}(a{x}^{ϵ}+b{y}^{ϵ}+c{z}^{ϵ}-\lambda t^{ϵ}).$$

(25)

By using Equation (25) in Equation (24), we get

$$\lambda W^{\prime }+a^2{W}^{\prime \prime }+b^2{W}^{\prime \prime }+c^2{W}^{\prime \prime }+aW{W}^{\prime }=0.$$

(26)

Taking the integration of Equation (26) and assuming the integration constant is 0, then we obtain

$$\lambda W+a^2{W}^{\prime }+b^2{W}^{\prime }+c^2{W}^{\prime }+a{\displaystyle \frac{W^2}{2}}=0.$$

(27)

By using the homogeneous balance method and balancing the terms $W^{\prime }$ and $W^2$, we get $m=1$.

3.1. Exact Solitons by ${exp}_a$ Function Method

Equation (5) changes to the below form for m = 1:

$$H\left(\xi \right)={\displaystyle \frac{{\alpha }_0+{\alpha }_1{d}^{\xi }}{{\beta }_0+{\beta }_1{d}^{\xi }}}.$$

(28)

Putting Equation (28) in Equation (27), a system is achieved. By solving, we obtain the following.

Solution set 1:

$$\left\{{\alpha }_0=0,{\beta }_1={\displaystyle \frac{a{\alpha }_1}{2log\left(d\right)\left(a^2+b^2+c^2\right)}},\lambda =-\left(a^2+b^2+c^2\right)log\left(d\right)\right\}.$$

(29)

$$w(x,y,z,t)={\displaystyle \frac{{\alpha }_1{d}^{\mathsf{\Lambda }}}{\frac{\left(a{\alpha }_1\right)d^{\mathsf{\Lambda }}}{2log\left(d\right)\left(a^2+b^2+c^2\right)}+{\beta }_0}}.$$

(30)

where $\mathsf{\Lambda }={\displaystyle \frac{\mathsf{\Gamma }(1+\varrho )}{ϵ}}(a{x}^{ϵ}+b{y}^{ϵ}+c{z}^{ϵ}+\left(a^2+b^2+c^2\right)log\left(d\right)t^{ϵ})$.

Solution set 2:

$$\left\{{\alpha }_1=0,{\beta }_0=-{\displaystyle \frac{a{\alpha }_0}{2log\left(d\right)\left(a^2+b^2+c^2\right)}},\lambda =\left(a^2+b^2+c^2\right)log\left(d\right)\right\}.$$

(31)

$$w(x,y,z,t)={\displaystyle \frac{{\alpha }_0}{{\beta }_1{d}^{\mathsf{\Lambda }}-\frac{a{\alpha }_0}{2log\left(d\right)\left(a^2+b^2+c^2\right)}}}.$$

(32)

where $\mathsf{\Lambda }={\displaystyle \frac{\mathsf{\Gamma }(1+\varrho )}{ϵ}}(a{x}^{ϵ}+b{y}^{ϵ}+c{z}^{ϵ}-\left(a^2+b^2+c^2\right)log\left(d\right)t^{ϵ})$.

3.2. Exact Wave Solutions by MSE Method

Equation (11) reduces for m = 1:

$$W\left(\mathsf{\Lambda }\right)=b_0+b_1\varphi \left(\mathsf{\Lambda }\right).$$

(33)

Inserting Equation (33) in Equation (27) along Equation (12), we achieve the following.

Solution set:

$$\left\{b_0=\mp {\displaystyle \frac{2i\sqrt{\eta }\left(a^2+b^2+c^2\right)}{a}},b_1=-{\displaystyle \frac{2\left(a^2+b^2+c^2\right)}{a}},\lambda =\pm 2i\sqrt{\eta }\left(a^2+b^2+c^2\right)\right\}.$$

(34)

Case 1:

$$w(x,y,z,t)={\displaystyle \frac{2\left(a^2+b^2+c^2\right)}{a}}(\mp i\sqrt{\eta }+\sqrt{-\eta }tanh\left(\sqrt{-\eta }{\displaystyle \frac{\mathsf{\Gamma }(1+\varrho )}{ϵ}}(a{x}^{ϵ}+b{y}^{ϵ}+c{z}^{ϵ}-\lambda t^{ϵ})\right)).$$

(35)

$$w(x,y,z,t)={\displaystyle \frac{2\left(a^2+b^2+c^2\right)}{a}}(\mp i\sqrt{\eta }+\sqrt{-\eta }coth\left(\sqrt{-\eta }{\displaystyle \frac{\mathsf{\Gamma }(1+\varrho )}{ϵ}}(a{x}^{ϵ}+b{y}^{ϵ}+c{z}^{ϵ}-\lambda t^{ϵ})\right)).$$

(36)

$$\begin{array}{cc}\hfill & w(x,y,z,t)={\displaystyle \frac{2\left(a^2+b^2+c^2\right)}{a}}(\mp i\sqrt{\eta }-(\sqrt{-\eta }\left(\right(-tanh\left(2\sqrt{-\eta }{\displaystyle \frac{\mathsf{\Gamma }(1+\varrho )}{ϵ}}(a{x}^{ϵ}+b{y}^{ϵ}+c{z}^{ϵ}-\lambda t^{ϵ})\right)\hfill \\ \hfill & \pm isech\left(2\sqrt{-\eta }{\displaystyle \frac{\mathsf{\Gamma }(1+\varrho )}{ϵ}}(a{x}^{ϵ}+b{y}^{ϵ}+c{z}^{ϵ}-\lambda t^{ϵ})\right)\left)\right)\left)\right).\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill & w(x,y,z,t)={\displaystyle \frac{2\left(a^2+b^2+c^2\right)}{a}}(\mp i\sqrt{\eta }-(\sqrt{-\eta }\left(\right(-coth\left(2\sqrt{-\eta }{\displaystyle \frac{\mathsf{\Gamma }(1+\varrho )}{ϵ}}(a{x}^{ϵ}+b{y}^{ϵ}+c{z}^{ϵ}-\lambda t^{ϵ})\right)\hfill \\ \hfill & \pm csch\left(2\sqrt{-\eta }{\displaystyle \frac{\mathsf{\Gamma }(1+\varrho )}{ϵ}}(a{x}^{ϵ}+b{y}^{ϵ}+c{z}^{ϵ}-\lambda t^{ϵ})\right)\left)\right)\left)\right).\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill & w(x,y,z,t)={\displaystyle \frac{2\left(a^2+b^2+c^2\right)}{a}}(\mp i\sqrt{\eta }-(-{\displaystyle \frac{\sqrt{-\eta }}{2}}\left(\right(tanh\left({\displaystyle \frac{\sqrt{-\eta }}{2}}{\displaystyle \frac{\mathsf{\Gamma }(1+\varrho )}{ϵ}}(a{x}^{ϵ}+b{y}^{ϵ}+c{z}^{ϵ}-\lambda t^{ϵ})\right)\hfill \\ \hfill & +coth\left({\displaystyle \frac{\sqrt{-\eta }}{2}}{\displaystyle \frac{\mathsf{\Gamma }(1+\varrho )}{ϵ}}(a{x}^{ϵ}+b{y}^{ϵ}+c{z}^{ϵ}-\lambda t^{ϵ})\right)\left)\right)\left)\right).\hfill \end{array}$$

(39)

Case 2:

$$w(x,y,z,t)={\displaystyle \frac{2\left(a^2+b^2+c^2\right)}{a}}(\mp i\sqrt{\eta }-\sqrt{\eta }tan\left(\sqrt{\eta }{\displaystyle \frac{\mathsf{\Gamma }(1+\varrho )}{ϵ}}(a{x}^{ϵ}+b{y}^{ϵ}+c{z}^{ϵ}-\lambda t^{ϵ})\right)).$$

(40)

$$w(x,y,z,t)={\displaystyle \frac{2\left(a^2+b^2+c^2\right)}{a}}(\mp i\sqrt{\eta }+\sqrt{\eta }cot\left(\sqrt{\eta }{\displaystyle \frac{\mathsf{\Gamma }(1+\varrho )}{ϵ}}(a{x}^{ϵ}+b{y}^{ϵ}+c{z}^{ϵ}-\lambda t^{ϵ})\right)).$$

(41)

$$\begin{array}{cc}\hfill & w(x,y,z,t)={\displaystyle \frac{2\left(a^2+b^2+c^2\right)}{a}}(\mp i\sqrt{\eta }-(\sqrt{\eta }\left(\right(tan\left(2\sqrt{\eta }{\displaystyle \frac{\mathsf{\Gamma }(1+\varrho )}{ϵ}}(a{x}^{ϵ}+b{y}^{ϵ}+c{z}^{ϵ}-\lambda t^{ϵ})\right)\hfill \\ \hfill & \pm sec\left(2\sqrt{\eta }{\displaystyle \frac{\mathsf{\Gamma }(1+\varrho )}{ϵ}}(a{x}^{ϵ}+b{y}^{ϵ}+c{z}^{ϵ}-\lambda t^{ϵ})\right)\left)\right)\left)\right).\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill & w(x,y,z,t)={\displaystyle \frac{2\left(a^2+b^2+c^2\right)}{a}}(\mp i\sqrt{\eta }-(\sqrt{\eta }\left(\right(-cot\left(2\sqrt{\eta }{\displaystyle \frac{\mathsf{\Gamma }(1+\varrho )}{ϵ}}(a{x}^{ϵ}+b{y}^{ϵ}+c{z}^{ϵ}-\lambda t^{ϵ})\right)\hfill \\ \hfill & \pm csc\left(2\sqrt{\eta }{\displaystyle \frac{\mathsf{\Gamma }(1+\varrho )}{ϵ}}(a{x}^{ϵ}+b{y}^{ϵ}+c{z}^{ϵ}-\lambda t^{ϵ})\right)\left)\right)\left)\right).\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill & w(x,y,z,t)={\displaystyle \frac{2\left(a^2+b^2+c^2\right)}{a}}(\mp i\sqrt{\eta }-({\displaystyle \frac{\sqrt{\eta }}{2}}\left(\right(tan\left({\displaystyle \frac{\sqrt{\eta }}{2}}{\displaystyle \frac{\mathsf{\Gamma }(1+\varrho )}{ϵ}}(a{x}^{ϵ}+b{y}^{ϵ}+c{z}^{ϵ}-\lambda t^{ϵ})\right)\hfill \\ \hfill & -cot\left({\displaystyle \frac{\sqrt{\eta }}{2}}{\displaystyle \frac{\mathsf{\Gamma }(1+\varrho )}{ϵ}}(a{x}^{ϵ}+b{y}^{ϵ}+c{z}^{ϵ}-\lambda t^{ϵ})\right)\left)\right)\left)\right).\hfill \end{array}$$

(44)

4. Graphical Interpretation

The graphical interpretation for the obtained solutions for the truncated M-fractional Bateman–Burgers model will be explained in this section.

In Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6, the authors use contour, three-dimensional, and two-dimensional plots to illustrate the results. Graphs for $ϵ=0.4,0.8,1$ are also used to illustrate the effect of fractional derivative.

5. Modulation Instability (MI) Analysis

Modulation instability is a phenomenon where a small perturbation in a continuous wave grows exponentially in time or space, leading to the breakdown of the continuous wave into a train of pulses or solitons. A steady-state method is examined for solving the Bateman–Burgers model, shown in [32,33]

$$w(x,y,z,t)=\left(W(x,y,z,t)+\sqrt{\tau }\right)e^{\iota \tau t}.$$

(45)

where $\tau$ shows optical power of normalizing.

Inserting Equation (45) into Equation (1). Using linearity, one gets

$$\iota {\tau }^{3/2}+W_t+\iota \tau W-W_{\mathrm{xx}}-W_{\mathrm{yy}}-W_{\mathrm{zz}}=0.$$

(46)

Supposing the solution of Equation (46) is mentioned as

$$W(x,y,z,t)=W_1{e}^{\iota (\rho x+\theta y+\kappa z-\omega t)}+W_2{e}^{-\iota (\rho x+\theta y+\kappa z-\omega t)}.$$

(47)

Here, $\rho$, $\theta$, $\kappa$, and $\omega$ are the constants. Equation (47) is substituted into Equation (46). After solving the determinant of the coefficient matrix, we obtain the dispersion relation by adding the coefficients of $e^{\iota (\rho x+\theta y+\kappa z-\omega t)}$ and $e^{-\iota (\rho x+\theta y+\kappa z-\omega t)}$:

$${\theta }^2{\kappa }^2+{\theta }^2{\kappa }^2+{\theta }^2{\rho }^2+{\theta }^2{\rho }^2+{\kappa }^4+2{\kappa }^2{\rho }^2+{\rho }^4-{\tau }^2+{\theta }^4+{\omega }^2=0.$$

(48)

The dispersion relation can be found from Equation (48) for q results

$$\omega =\pm \sqrt{-{\theta }^4-2{\theta }^2{\kappa }^2-2{\theta }^2{\rho }^2-{\kappa }^4-2{\kappa }^2{\rho }^2-{\rho }^4+{\tau }^2}.$$

(49)

The steady-state stability is demonstrated by the gained dispersion relation. The steady-state outcome will not be stable when wave number $\omega$ is not real because the perturbation increases exponentially. However, a steady state becomes stable against minor perturbations if $\omega$ is not imaginary. The result in the unstable steady state is

$$-{\theta }^4-2{\theta }^2{\kappa }^2-2{\theta }^2{\rho }^2-{\kappa }^4-2{\kappa }^2{\rho }^2-{\rho }^4+{\tau }^2<0.$$

(50)

It is possible to acquire the MI gain spectrum $G\left(p\right)$ as follows:

$$G\left(\rho \right)=2Im\left(\omega \right)=\pm \sqrt{-{\theta }^4-2{\theta }^2{\kappa }^2-2{\theta }^2{\rho }^2-{\kappa }^4-2{\kappa }^2{\rho }^2-{\rho }^4+{\tau }^2}.$$

(51)

6. Conclusions

The authors have succeeded in achieving the exact solitons to the truncated M-fractional Bateman–Burgers equation by applying the two methods named as the ${exp}_a$ function method and the modified simplest equation method. We use Mathematica software to obtain and to verify solutions. The solutions obtained do not exist in the literature. Some gained results are also represented by two-dimensional (2D), three-dimensional (3D), and contour graphs with the help of the Mathematica tool. In addition, the stationary results of the governing equation are studied through modulation instability. Graphical explanation of modulation instability analysis is given in Figure 7. It is suggested that the techniques employed have applicability for other nonlinear equations in different scientific and engineering domains. The attained solutions may be helpful for the development of the governing equation and other related fields, including fluid dynamics, nonlinear acoustics, aerodynamics, shock wave propagation, weather forecasting, oceanography, combustion dynamics, etc., in future research. Moreover, new models can be derived by taking some specific sets of parameters from this model. We plan to analyze these models in future studies.