Lump-Type Solutions, Mixed Solutions and Rogue Waves for a (3+1)-Dimensional Variable-Coefficients Burgers Equation

Abstract

In this work, we consider the (3+1)-dimensional Burgers equation with variable coefficients, which is frequently used to define the motion of solitary waves. Abundant lump waves are constructed by taking the ansatz as a rational function. Furthermore, mixed solutions utilizing lump waves, rogue waves, and kink solitons are obtained by combining the rational function with an exponential function, resulting in fission and fusion phenomena.

1. Introduction

Nonlinear evolution equations (NLEEs) have gained increasing attention in the field of nonlinear science due to their many applications in various fields, such as fluid mechanics [1,2], plasma physics [3], optics [4], biology [5], and condensed matter physics [6]. The most important of these studies are meant to obtain soliton solutions. Many methods have been developed, including the inverse scattering transformation [7,8], the Darboux transformation [9], Bäcklund transformations [10], and Hirota’s bilinear technique [11].

The lump waves in the Kadomtsev–Petviashvili (KP) equation were theoretically predicted using the long wave limit approach [12]. This type of solution was observed in optical fibers [13], which attracted the attention of researchers. Then, a symbolic computation approach was proposed to construct additional lump wave solutions of NLEEs, which was successfully used in the KP equation [14], the BKP-Korteweg de-Vries (BKP-KdV) equation [15], the KP3-4 equation [16], the higher-order KdV-5 equation [17], and many more [18,19,20,21]. Furthermore, M-lump solutions were constructed by making some parameters conjugate to each other [22]. The mixed lump wave solutions of the (2+1)-dimensional Sawada–Kotera equation were obtained by an ingenious limit approach [23]. In addition, interesting research on lump waves, higher-order lump waves, and the interaction between waves was conducted using the ansatz method, such as the spatial symmetric (2+1)-dimensional dispersive wave model [24] and the (2+1)-dimensional Mel’nikov equation [25]. Compared to the version of NLEEs with constant coefficients, the one with variable coefficients can obtain a larger number of new solutions for complex physical models and describe realistic physical phenomena [26]. An example of this is the variable-coefficients, high-order, nonlinear Schrödinger equation (VcHNLS). By using reductive perturbation theory with long wave approximation, VcHNLS can be used to study the propagation of pulse waves in deformable elastic vessels filled with inviscid blood [27].

Burgers-type equations are frequently used to define the motion of solitary waves in many fields, such as ocean dynamics [28], vehicular traffic [29], and particle systems [30]. Recently, more exact solutions to Burgers-type equations have been constructed by various methods, such as the nonlocal symmetry group reduction method [31], the consistent tanh expansion method [32], and the velocity resonance method [33]. In Ref. [34], soliton-like and period-form solutions were gained using the extended tanh method. Based on the ansatz method, lump solutions and period lump solutions were constructed in Ref. [35]. Two types of semi-rational solutions were unearthed for the constant-coefficient (3+1)-dimensional Burgers equation [36]. By means of the test function method, lump waves and mixed solutions were constructed for the generalized (2+1) Burgers equation with variable coefficients [37]. In addition, by using white noise analysis, Hermite transformation of the solitary wave solutions of the Wick-type stochastic Burgers equation (WSB) were discussed, which evolved the density field of the weakly asymmetric exclusion process. The WSB is of the form

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill U_t& +p\left(t\right)(U_{xx}+U♢U_x)=W\left(t\right)♢R^{♢}(t,U,U_x,U_{xx}),\hfill \end{array}\end{array}$$

(1)

which is the perturbation of the Burgers equation with variable coefficients

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u_t& +p\left(t\right)(u_{xx}+u{u}_x)=0.\hfill \end{array}\end{array}$$

(2)

Here, $p\left(t\right)$ is a function of t, $W\left(t\right)$ defines the Gaussian white noise, and “♢” denotes the Wick production in the Kondratiev distribution space [38]. Motivated by the above scientific applications, the aim of this paper is to investigate additional new solutions to a (3+1)-dimensional variable-coefficients Burgers equation (VcB):

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u_y& +m\left(t\right)u_x+n\left(t\right)u_z+p\left(t\right)(u_{xx}+2v{u}_x)\hfill \\ & +q\left(t\right)(u_{yy}+2u{u}_y)+r\left(t\right)(u_{zz}+2w{u}_z)+s\left(t\right)u_t=0,\hfill \\ & u_x=v_y,u_z=w_y,\hfill \end{array}\end{array}$$

(3)

with real functions $u=u(x,y,z,t)$, $v=v(x,y,z,t)$, and $w=w(x,y,z,t)$, while m, n, p, q, r, and s are any real functions with respect to time-independent variable t. However, mixed lump waves and solitons and rogue waves of VcB (3) have not yet been found.

The structure of this paper is as follows: Rational solutions are found by means of the ansatz technique in Section 2. Section 3 is aimed at finding solutions to the interaction between lump waves and multi-kink solitons. The details of the effects of the ansatz function are provided, and some special types of interactions are unearthed, such as fission, fusion, and rogue wave phenomena. Section 4 consists of discussion, some features, and comments. It should be noted that the graphical representation of the solution was provided by Maple.

2. Rational Solution of the (3+1)-Dimensional Variable-Coefficients Burgers Equation

An effective method for solving a PDE is to use the Hirota method [11]. In this section, by means of this method, we seek the rational solution for VcB (3), and the first thing to do is to use the following dependent variable transformation:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill u={(lnf)}_y,v={(lnf)}_x,w={(lnf)}_z,\end{array}\end{array}$$

(4)

and to rewrite VcB (3) in the following bilinear form:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & f_y^2-f{f}_{yy}+m\left(t\right)(f{f}_{xy}-f_x{f}_y)+n\left(t\right)(f{f}_{yz}-f_y{f}_z)+p\left(t\right)(f{f}_{xxy}-f_{xx}{f}_y)\hfill \\ & +q\left(t\right)(f{f}_{yyy}-f_y{f}_{yy})+r\left(t\right)(f{f}_{yzz}-f_y{f}_{zz})+s\left(t\right)(f_y{f}_t-f{f}_{yt})=0,\hfill \end{array}\end{array}$$

(5)

where f is a real function. In order to obtain the rational solution of VcB (3), we assume that f has the following form:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill f={\xi }^2+{\varphi }^2+a_9,\end{array}\end{array}$$

(6)

with $\xi (x,y,z,t)=a_{1}x+a_{2}y+a_{3}z+a_4\left(t\right)$ and $\varphi (x,y,z,t)=a_{5}x+a_{6}y+a_{7}z+a_8\left(t\right)$, where $a_4\left(t\right)$ and $a_8\left(t\right)$ are any real functions related to t and $a_i(i=1,2,\cdots ,9)$ are arbitrary real parameters. Substituting Equation (6) into Equation (5) and vanishing all the coefficients of x, y, z, and t, the relations among arbitrary constants are obtained as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p\left(t\right)& =-\frac{q\left(t\right)(a_2^2+a_6^2)}{a_1^2+a_5^2}-\frac{r\left(t\right)(a_3^2+a_7^2)}{a_1^2+a_5^2},\hfill \\ \hfill s\left(t\right)& =\frac{m\left(t\right)(a_1{a}_7-a_3{a}_5)+a_2{a}_7-a_6{a}_3}{a_7{a}_4{\left(t\right)}^{\prime }-a_3{a}_8{\left(t\right)}^{\prime }},\hfill \\ \hfill n\left(t\right)& =\frac{m\left(t\right)[a_1{a}_8{\left(t\right)}^{\prime }-a_5{a}_4{\left(t\right)}^{\prime }]+a_2{a}_8{\left(t\right)}^{\prime }-a_6{a}_4{\left(t\right)}^{\prime }}{a_7{a}_4{\left(t\right)}^{\prime }-a_3{a}_8{\left(t\right)}^{\prime }},\hfill \end{array}\end{array}$$

(7)

where $a_8{\left(t\right)}^{\prime }\equiv \frac{\mathrm{d}{a}_8\left(t\right)}{\mathrm{d}t}$ references the derivative of $a_8\left(t\right)$ about t. Substituting Equations (6) and (7) into the transformation $u={(lnf)}_y$, the exact solution is written as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill u={(lnf)}_y=2\frac{(a_1{a}_2+a_5{a}_6)x+(a_2^2+a_6^2)y+(a_2{a}_3+a_6{a}_7)z+a_4\left(t\right)a_2+a_8\left(t\right)a_6}{{[a_{1}x+a_{2}y+a_{3}z+a_4\left(t\right)]}^2+{[a_{5}x+a_{6}y+a_{7}z+a_8\left(t\right)]}^2+a_9}.\end{array}\end{array}$$

(8)

Here, we obtain two kinds of lump wave.

Example 1.

When $a_4\left(t\right)=t$, $a_8\left(t\right)=3t+4$, $m\left(t\right)=t^2$, $r\left(t\right)=-t$, $a_1=8$, $a_2=1$, $a_3=0.2$, $a_5=a_7=2$, $a_6=0.01$, and $a_9=4$, their spatial structure and the propagation of the lump of u are described in Figure 1 at $t=-25$, $t=0$, and $t=25$.

According to Figure 1, it can be found that the lump wave moves in the positive direction of the y axis and the negative direction of the z axis. Figure 1a is the three-dimensional image of the lump solution, which shows the spatial structure with a peak and a trough, and the solution decays in all spatial directions.

Example 2.

When $a_4\left(t\right)=e^{\frac{t}{4}}$, $a_8\left(t\right)=e^{\frac{t}{4}}$, $m\left(t\right)=-t$, $r\left(t\right)=t$, $a_1=a_6=1$, $a_2=4$, $a_3=a_5=a_7=2$, and $a_9=4$, the three-dimensional structure plots of the lump wave solution are obtained, and it is obvious that when $t<0$, the lump wave remains stationary. On the contrary, when $t>0$, the lump wave starts to move, and with the increase of t, the lump wave quickly moves away from the initial position. The lump wave solution is stable when it exists alone. Could the lump wave maintain its properties after colliding with solitons or other nonlinear waves? Next, we will discuss the interaction between solitons and lump waves below.

3. Interaction between Solitons and Lump Waves

In the following section, we focus on constructing the interaction solutions between lump waves and solitons by combining the rational functions and exponential functions. First of all, we take the function f as the ansatz with the following rational–exponential function:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill f={\xi }^2+{\varphi }^2+a_9+{\lambda }_1\left(t\right)e^{{\varphi }_{1}x+{\varphi }_{2}y+{\varphi }_{3}z+{\varphi }_4\left(t\right)}+{\lambda }_2\left(t\right)e^{-{\varphi }_{1}x-{\varphi }_{2}y-{\varphi }_{3}z-{\varphi }_4\left(t\right)},\end{array}\end{array}$$

(9)

with $\xi (x,y,z,t)=a_{1}x+a_{2}y+a_{3}z+a_4\left(t\right)$ and $\varphi (x,y,z,t)=a_{5}x+a_{6}y+a_{7}z+a_8\left(t\right)$, where $a_4\left(t\right)$, $a_8\left(t\right)$, and ${\varphi }_4\left(t\right)$ are any real functions related to t, and the parameters $a_i(i=1,2,\cdots ,9)$ and ${\varphi }_i(i=1,2,3)$ are determined real parameters. By substituting Equation (9) into Equation (5) and vanishing the different powers of the variables x, y, z, and t, we obtain a series of algebraic equations on the undetermined parameters. We will discuss two cases questioning whether ${\lambda }_1\left(t\right)$ is equal to zero, and the same number of cases questioning whether ${\lambda }_2\left(t\right)$ is zero is similar to that not discussed here.

Case 1: ${\lambda }_1\left(t\right)=0$

After solving these algebraic equations, a set of constraining relations is obtained as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill s\left(t\right)& =\frac{m\left(t\right)(a_3{a}_5-a_1{a}_7)+a_3{a}_6-a_2{a}_7}{a_3{a}_8{\left(t\right)}^{\prime }},q\left(t\right)=-\frac{(a_1^2+a_5^2)p\left(t\right)}{a_2^2+a_6^2}-\frac{(a_3^2+a_7^2)r\left(t\right)}{a_2^2+a_6^2},\hfill \\ \hfill {\lambda }_2\left(t\right)& =c_{1}exp\left[\int \frac{g_1}{(a_2^2+a_6^2)(a_1{a}_{7}m\left(t\right)-a_3{a}_{5}m\left(t\right)+a_2{a}_7-a_3{a}_6)}dt\right],\hfill \\ \hfill g_1& =(a_2^2+a_6^2)[(a_8{\left(t\right)}^{\prime }{\varphi }_1-a_5{\varphi }_4{\left(t\right)}^{\prime })a_3+(a_7{\varphi }_4{\left(t\right)}^{\prime }-a_8{\left(t\right)}^{\prime }{\varphi }_3)a_1]m\left(t\right)\hfill \\ & +a_3^3{a}_8{\left(t\right)}^{\prime }{\varphi }_2^{2}r\left(t\right)+[{\varphi }_2^2{a}_7^2{a}_8{\left(t\right)}^{\prime }r\left(t\right)+({\varphi }_2-{\varphi }_3^{2}r\left(t\right))a_2^2{a}_8{\left(t\right)}^{\prime }\hfill \\ & +({\varphi }_2-{\varphi }_3^{2}r\left(t\right))a_6^2{a}_8{\left(t\right)}^{\prime }]a_3+a_2(a_2^2+a_6^2)(a_7{\varphi }_4{\left(t\right)}^{\prime }-a_8{\left(t\right)}^{\prime }{\varphi }_3),\hfill \\ & -[{\varphi }_1^2(a_2^2+a_6^2)-{\varphi }_2^2(a_1^2+2a_5^2)]a_8{\left(t\right)}^{\prime }{a}_3,\hfill \\ \hfill n\left(t\right)& =-\frac{a_{1}m\left(t\right)+a_2}{a_3},{\lambda }_1\left(t\right)=0,a_4\left(t\right)=c_2,\hfill \end{array}\end{array}$$

(10)

with the integral constant $c_1$. The interaction solution between the lump wave and the soliton of the (3+1)-dimensional VcB is constructed:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill u=2{(lnf)}_y=\frac{4a_2\xi +4a_6\varphi -2{\varphi }_2{c}_1{l}_1}{{\xi }^2+{\varphi }^2+c_1{l}_1+a_9},\end{array}\end{array}$$

(11)

where

$$\begin{array}{c}\hfill \begin{array}{cc}& \xi =a_{1}x+a_{2}y+a_{3}z+c_2,\varphi =a_{5}x+a_{6}y+a_{7}z+a_8\left(t\right),\hfill \\ & l_1=exp\left[\int \frac{g_1}{(a_3^2+a_7^2)(a_1{a}_{7}m\left(t\right)-a_3{a}_{5}m\left(t\right)+a_2{a}_7-a_3{a}_6)}\mathrm{d}t\right]e^{-{\varphi }_{1}x-{\varphi }_{2}y-{\varphi }_{3}z-{\varphi }_4\left(t\right)}.\hfill \end{array}\end{array}$$

(12)

Example 3.

When the parameters are selected as

$$\begin{array}{c}\hfill \begin{array}{cc}& a_1=8,a_2=c_2={\varphi }_3=1,a_3=\frac{1}{5},a_5=a_7=c_1=2,a_6=\frac{1}{100},a_9=4,\hfill \\ & {\varphi }_1={\varphi }_2=\frac{1}{2},r\left(t\right)=t,a_8\left(t\right)=3t+3,m\left(t\right)=\frac{1}{2t^2},p\left(t\right)=\frac{1}{10},{\varphi }_4\left(t\right)=t.\hfill \end{array}\end{array}$$

(13)

The solution utilizing an interaction between a kink wave and a lump wave is obtained as shown in Figure 2. Figure 2a–c show the three-dimensional plot at $t=-50$, $t=0$, and $t=200$, respectively. According to the figures, it is found that the speed of the kink soliton is greater than that of the lump wave and that the kink soliton can catch up with the lump wave in this time period. In the process of interaction, the amplitude of the lump wave becomes smaller and smaller until the lump wave is completely engulfed and continues to spread. The convergence phenomenon of kink solitons and lump waves is observed.

Case 2: ${\lambda }_1\left(t\right)\ne 0$

By the same operation, the solution of the interaction between a soliton and a lump wave is obtained as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill u=2{(lnf)}_y=\frac{4a_2\xi +4a_6\varphi +2{\varphi }_2(c_1{l}_1-c_2{l}_2)}{{\xi }^2+{\varphi }^2+c_1{l}_1+a_9},\end{array}\end{array}$$

(14)

where

$$\begin{array}{c}\hfill \begin{array}{cc}& \xi =a_{1}x+a_{2}y+a_{3}z+a_4\left(t\right),\varphi =a_{5}x+a_{6}y+a_{7}z+a_8\left(t\right),\hfill \\ & l_1=exp\left[\int \frac{-g_1}{(a_3^2+a_7^2)(a_1{a}_{7}m\left(t\right)-a_3{a}_{5}m\left(t\right)+a_2{a}_7-a_3{a}_6)}\mathrm{d}t\right]e^{{\varphi }_{1}x+{\varphi }_{2}y+{\varphi }_{3}z+{\varphi }_4\left(t\right)},\hfill \\ & l_2=exp\left[\int \frac{g_2}{(a_3^2+a_7^2)(a_1{a}_{7}m\left(t\right)-a_3{a}_{5}m\left(t\right)+a_2{a}_7-a_3{a}_6)}\mathrm{d}t\right]e^{-{\varphi }_{1}x-{\varphi }_{2}y-{\varphi }_{3}z-{\varphi }_4\left(t\right)}.\hfill \end{array}\end{array}$$

(15)

The corresponding parameters satisfy the following requirements:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill s\left(t\right)& =\frac{m\left(t\right)(a_3{a}_5-a_1{a}_7)+a_3{a}_6-a_2{a}_7}{a_3{a}_8{\left(t\right)}^{\prime }},q\left(t\right)=-\frac{(a_1^2+a_5^2)p\left(t\right)}{a_2^2+a_6^2}-\frac{(a_3^2+a_7^2)r\left(t\right)}{a_2^2+a_6^2},\hfill \\ \hfill {\lambda }_1\left(t\right)& =c_{1}exp\left[\int \frac{-g_1}{(a_2^2+a_6^2)(a_1{a}_{7}m\left(t\right)-a_3{a}_{5}m\left(t\right)+a_2{a}_7-a_3{a}_6)}dt\right],\hfill \\ \hfill g_1& =(a_2^2+a_6^2)[a_3{\varphi }_1^{2}p\left(t\right)+a_3{\varphi }_3^{2}r\left(t\right)-a_1{\varphi }_{3}m\left(t\right)+a_3{\varphi }_{1}m\left(t\right)-a_2{\varphi }_3\hfill \\ & +a_3{\varphi }_2]a_8{\left(t\right)}^{\prime }-a_3{\varphi }_2^2[(a_1^2+a_5^2)p\left(t\right)+(a_3^{2}r+a_7^2)r\left(t\right)]a_8{\left(t\right)}^{\prime }\hfill \\ & +(a_2^2+a_6^2)[a_1{a}_{7}m\left(t\right)-a_3{a}_{5}m\left(t\right)+a_2{a}_7-a_3{a}_6]{\varphi }_4{\left(t\right)}^{\prime },\hfill \\ \hfill {\lambda }_2\left(t\right)& =c_{2}exp\left[\int \frac{g_2}{(a_2^2+a_6^2)(a_1{a}_{7}m\left(t\right)-a_3{a}_{5}m\left(t\right)+a_2{a}_7-a_3{a}_6)}dt\right],\hfill \\ \hfill g_2& =-(a_2^2+a_6^2)[a_3{\varphi }_1^{2}p\left(t\right)+a_3{\varphi }_3^{2}r\left(t\right)+a_1{\varphi }_{3}m\left(t\right)-a_3{\varphi }_{1}m\left(t\right)+a_2{\varphi }_3\hfill \\ & -a_3{\varphi }_2]a_8{\left(t\right)}^{\prime }+a_3{\varphi }_2^2[(a_1^2+a_5^2)p\left(t\right)+(a_3^{2}r+a_7^2)r\left(t\right)]a_8{\left(t\right)}^{\prime }\hfill \\ & +(a_2^2+a_6^2)[a_1{a}_{7}m\left(t\right)-a_3{a}_{5}m\left(t\right)+a_2{a}_7-a_3{a}_6]{\varphi }_4{\left(t\right)}^{\prime },\hfill \\ \hfill n\left(t\right)& =-\frac{a_{1}m\left(t\right)+a_2}{a_3},a_4\left(t\right)=c_3.\hfill \end{array}\end{array}$$

(16)

Here, we obtain two different forms by taking $m\left(t\right)$ as $\frac{2}{t^2}$ and $e^t$, and when $m\left(t\right)$ is taken to the non-even power of t, the solution will have singularity, which will not be discussed.

Example 4.

The appropriate parameters are selected as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill a_3& =a_5=a_7=c_1=c_3=p\left(t\right)=2,a_1=8,{\varphi }_1={\varphi }_2=\frac{1}{2},c_2={\varphi }_3=1,\hfill \\ \hfill a_2& =a_9=4,a_6=0.01,a_8\left(t\right)=3t+4,q\left(t\right)=r\left(t\right)={\varphi }_4\left(t\right)=t,m\left(t\right)=\frac{2}{t^2}.\hfill \end{array}\end{array}$$

(17)

Then, the corresponding interaction solution between two kink solitons and a lump wave can be described by Figure 3a–e. We find that the two kink waves keep moving in opposite directions along the z axis. When $t<0$, the lump wave appears between the two kink solitons, and the two kink solitons gradually swallow the lump wave until there are only two kink solitons in the space at $t>0$. Finally, the lump wave gradually appears between the two kink waves. The density plot in Figure 3f shows that the lump wave disappears completely between the two kink solitons at $t=0$. After the interaction between the lump wave and the soliton, the amplitudes, shapes, and velocities remain unchanged.

Example 5.

We select the parameters as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill a_3& =a_5=a_7=c_1=c_3=p\left(t\right)=2,a_1=8,{\varphi }_1={\varphi }_2=\frac{1}{2},c_2={\varphi }_3=1,\hfill \\ \hfill a_2& =a_9=4,a_6=-1,a_8\left(t\right)=3t+4,q\left(t\right)=r\left(t\right)={\varphi }_4\left(t\right)=t,m\left(t\right)=e^t.\hfill \end{array}\end{array}$$

(18)

The interaction solution for two kink solitons and a lump wave is obtained, and the dynamic behavior of the solution is shown in Figure 4. As shown in Figure 4a–c, the three-dimensional plots of the interaction between two kink solitons and a lump wave are presented at $x=-10$, $x=0$, and $x=30$, respectively. It can be found that two kink waves move in the same direction with the same velocity along the negative direction of the z axis. When $x<0$, the lump wave gradually appears between the two kink waves, and when $x>0$, the two kink solitons gradually swallow the lump wave until there are only two kink solitons in the space, that is, the phenomenon of splitting occurs first, followed by converging. The phenomenon of the splitting and then polymerizing of two solitons and a lump wave also exists in the (3+1)-dimensional constant coefficient Burgers equation [35] and other (3+1)-dimensional nonlinear systems [39]. For the variable-coefficients equation, when the appropriate parameters are utilized, the subsequent phenomena of first the convergence and then the splitting of two solitons and a lump wave can be obtained.

Example 6.

In this section, we attempt to reveal the rogue wave excited by the two kink solitons for a VcB (3). Let

$$\begin{array}{c}\hfill \begin{array}{c}\hfill u=2{(lnf)}_y=\frac{4a_2\xi +4a_6\varphi +2{\varphi }_2(c_1{l}_1-c_2{l}_2)}{{\xi }^2+{\varphi }^2+c_1{l}_1+a_9},\end{array}\end{array}$$

(19)

where

$$\begin{array}{c}\hfill \begin{array}{cc}& \xi =a_{1}x+a_{2}y+a_{3}z+c_3,\varphi =a_{5}x+a_{6}y+a_{7}z+a_8\left(t\right),\hfill \\ & l_1=exp\left[\int \frac{-g_1}{(a_3^2+a_7^2)(a_1{a}_{7}m\left(t\right)-a_3{a}_{5}m\left(t\right)+a_2{a}_7-a_3{a}_6)}\mathrm{d}t\right]e^{{\varphi }_{1}x+{\varphi }_{2}y+{\varphi }_{3}z+{\varphi }_4\left(t\right)},\hfill \\ & l_2=exp\left[\int \frac{g_2}{(a_3^2+a_7^2)(a_1{a}_{7}m\left(t\right)-a_3{a}_{5}m\left(t\right)+a_2{a}_7-a_3{a}_6)}\mathrm{d}t\right]e^{-{\varphi }_{1}x-{\varphi }_{2}y-{\varphi }_{3}z-{\varphi }_4\left(t\right)}.\hfill \end{array}\end{array}$$

(20)

The corresponding parameters satisfy the following requirements:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill a_3& =a_5=a_7=c_1=c_3=p\left(t\right)=2,a_1=8,{\varphi }_1={\varphi }_2=\frac{1}{2},c_2={\varphi }_3=1,\hfill \\ \hfill a_2& =a_9=4,a_6=-1,a_8\left(t\right)=sin\left(t\right)+4,q\left(t\right)=r\left(t\right)={\varphi }_4\left(t\right)=t,m\left(t\right)=t^2+6.\hfill \end{array}\end{array}$$

(21)

As seen in Figure 5, Equation (19) is a rogue wave excited by two kink solitons. Figure 5d shows that a rogue wave generates at $x=0$, and the middle fold of the kink soliton becomes higher than before.

4. Discussion

In this paper, a (3+1)-dimensional variable-coefficients Burgers Equation (3) has been investigated, evolving from a bilinear form. Rational and rational–exponential solutions have been constructed via the direct ansatz method. By choosing the arbitrary functions suitably, the solutions involving rational and rational–exponential functions become lump waves and lump–kink waves, respectively. Based on a deep study, the lump solution driven by the bilinear form is a stable peak as the time increases, and the fission and fusion phenomena between soliton and lump waves have been discussed. Moreover, we attained a lump wave excited by two kink solitons. Finally, our study is not restricted to water rogue waves [40,41], but is also evident among rogue waves excited by two kink solitons. It may be applied to hydrodynamics and particle systems.