Wave Solutions for a (2 + 1)-Dimensional Burgers–KdV Equation with Variable Coefficients via the Functional Expansion Method

Abstract

A (2 + 1)-dimensional fourth order Burgers–KdV equation with variable coefficients (vcBKdV) is studied here and interesting wave-type solutions with variable amplitudes and velocities are reported. The model has been not previously studied in the chosen form and it presents a twofold interest: as a model describing a rich variety of phenomena and as a higher-order equation solving difficulties generated by the presence of the variable coefficients. The novelty of our approach is related to the use of the functional expansion, a solving method based on an auxiliary equation that generalizes other approaches, such as, for example, the $\left(\frac{G^{\prime }}{G}\right)$ proved here. We use a similarity reduction with a nonlinear wave variable that leads to a determining system that it is not usually algebraic, but an over-determined system of partial differential equations. It depends on 14 constant or functional parameters and can generate much richer classes of solutions. Three such classes of solutions, corresponding to the case when a specific form of the generalized reaction–diffusion equation is used as auxiliary equation, are considered. The influence on the dynamical behavior of two important factors, the choices of the auxiliary equation and the form of solution, are studied by providing graphical representations of specific solutions for various values of the parameters.

1. Introduction

In many studies of nonlinear science, nonlinear partial differential equations (NPDEs) are employed to model important phenomena, with applications in many fields, ranging from fluid dynamics, nonlinear optics, plasma physics, or gravity to differential and algebraic geometry, topology and so on. Obtaining as many classes of solutions as possible for such NPDEs is of great interest in the current literature. New solving methods or modifications to the existing ones have been proposed and applied for this purpose. Part of them, as the inverse scattering [1], the Lax pair [2], the Hirota bilinear transformation [3,4,5,6] allows us to decide on the integrability and to obtain general classes of solutions. However, when the Cauchy problem for such equations cannot be explicitly solved, there are approaches that allow us to obtain specific classes of solutions, like, for example, wave-type solutions. These solutions enable the exploration of various significant properties of the physical phenomena and they can be obtained by applying various approaches. We can use the symmetry properties of the model, proceeding either to a dimensional reduction, as in the case of Lie [7,8,9,10] and of the generalized conditional symmetries [11,12], or to an extension of space, as in the case of Becchi–Rouet–Stora–Tyutin (BRST) symmetry [13,14,15,16]. Among the proposed approaches that are very efficient for geting wave solutions of NPDEs, we can mention: various extended tanh methods [17,18], the soliton ansatz method [19,20,21], the Jacobi elliptic method [22], the generalized Riccati equation method [23], and the $\left(\frac{G^{\prime }}{G}\right)$-expansion method [24,25]. The last three are the most common techniques that belong to the solving methods based on an auxiliary equation. We will apply, here, another approach called the functional expansion method, as proposed in [26].

Due to the increased importance of the inhomogeneous media with non-uniform boundaries, an increased interest has appeared, in recent times, in studying nonlinear partial differential equations with variable coefficients [27,28,29,30]. This option is motivated by the fact that, despite the increased difficulty, these equations can describe more complex and realistic phenomena than their counterparts with constant coefficients [31,32,33]. In non-autonomous media presented in various theoretical and experimental studies, an important task is that of detecting soliton-type solutions with time-varying velocities and amplitudes [34,35,36]. For the equation with variable coefficients, this task requires special care. It has been shown that, in some cases, it is possible to associate to each such equation an equation with constant coefficients [37]. To find solutions to the latter equation, the most convenient approach is to reduce the considered NPDE to a nonlinear ordinary differential equation (NODE). The Lie symmetry group [7,38,39] or nonclassical symmetry [40] methods are the most convenient approach to similarity reduction. A direct method was proposed by Clarkson and Kruskal [41]. When the reduction of non-autonomous NPDEs leads to NODEs, the determining systems take the form of algebraic equations, but if this reduction cannot be carried out directly, other approaches must be considered. The determining systems could, in these cases, become PDE systems and how they could be obtained is also a very important and theoretically challenging problem. Motivated by the reasons above, we will investigate the following $(2+1)$-variable coefficients Burgers–KdV equation:

$$u_{xt}+a\left(t\right)u_{4x}+b\left(t\right)u_{3x}+c\left(t\right)u_x^2+f\left(t\right)u{u}_{2x}+g\left(t\right)u_{2y}=0.$$

(1)

Here, $u(t,x,y)$ is the wave-amplitude function, $a\left(t\right),$ $b\left(t\right),$ $g\left(t\right)$ are the fourth-, third- and second-order spatial dispersive coefficients, respectively, and $c\left(t\right)$ is the nonlinear coefficient, while $f\left(t\right)$ represents the dissipative coefficient. Equation (1) describes the various physical models widely used in solid-state materials, plasmas, fluids and nonlinear optics [42,43]. A similar equation with constant coefficients was considered in [44] and its choice is justified by the fact that it was studied in [45] using the Lie symmetry group method, highlighting possible similarity reductions and corresponding solutions. Because of the variable coefficients now considered, the solving procedure requires more care, by choosing, for example, the modified version proposed in [46] instead of the Clarkson–Kruskal direct solution method.

In this paper, we will consider a reduction of the Equation (1) along a similarity variable $\xi$ linear in x but that could be nonlinear in y and t. We will prove that Equation (1) admits solutions compatible with an auxiliary equation of the reaction–diffusion type; that is, it can describe physical phenomena related to reaction, convection and diffusion. A more complicated dynamic system will be generated in the non-autonomous case. Unlike the case of autonomous systems, the determining system will now not be a system of algebraic equations, but one with differential equations. This determining system will be solved using the functional expansion method that will be briefly presented below.

The autonomous or non-autonomous Burgers–KdV equations arise from various contexts in physics, as nonlinear models incorporating the effects of dispersion, dissipation and nonlinearity. Many physical problems can be described by these dynamical equations. Typical examples are provided by the behaviour of long waves in shallow water and waves in plasmas [47], the flow of liquids containing gas bubbles [48] and the propagation of waves in an elastic tube field with a viscous fluid [49]. Although there are a lot of studies for the standard Burgers–KdV equation, with a lot of profound results [50,51], it seems that studying the vcBKdV in higher dimensions is an open and interesting problem [52]. As far as we know, only $(1+1)$-dimensional non-autonomous models have been considered up until now [53,54]. In this study, we take into consideration the problem of searching for multiple explicit wave solutions for the $(2+1)$-dimensional vcBKdV given by (1) via a generalized expansion method. The basics of this approach are outlined in Section 2, while its usefulness is proven for the ($2+1$)-dimensional vcBKdV in Section 3. Families of solutions with multiple parameters are pointed out according to the respective constraints between several parameters and by using the solutions of a generalized reaction–diffusion equation. The influence of the expressions taken by the variable coefficients upon these solutions is discussed as well. Some representative wave solutions are depicted. To the best of our knowledge, the structure of wave solutions related to this ($2+1$)-dimensional non-autonomous model is presented here for the first time. Section 4 is dedicated to a discussion of the results, while Section 5 covers the main conclusions of our study.

2. Basics on the Solving Method

2.1. Description of the Functional Expansion Method

The functional expansion method is an interesting version of the auxiliary equation method that i very often used for finding classes of solutions for important NPDEs. Essentially, the auxiliary equation method involves two steps: (i) using the symmetry properties of the NPDE, one proceeds from its “similarity reduction” to a nonlinear ordinary differential equation; and (ii) searching for NODEs solutions as combinations of known solutions of what we will choose as the auxiliary equation. The first step is implemented by a change of variables, from considered space-time variables, let us say {$x,y,t$}, to a single variable $\xi =\xi (x,y,t)$ known as the “wave variable”. The second step supposes to look for solutions of the NODE obtained in the previous step as expansions in terms of the known solutions, $G\left(\xi \right)$, and of their derivatives, $G^{\prime }$, of an auxiliary equation. In the case of the functional expansion method, solutions are sought in the specific form mentioned in [26].

To be very specific, let us consider a NPDE with three independent variables, $x,$ y and t, given by

$$E(t,u,u_t,u_x,u_y,u_{tt},u_{tx},u_{ty},u_{xy},u_{xx},u_{yy},\dots )=0,$$

(2)

where $u=u(t,x,y)$ is an unknown physical field and E denotes a polynomial along with its arguments. The functional expansion method allows us to find wave solutions of (2) by transforming it into a nonlinear ordinary differential equation (NODE) with the help of the wave variable, and, then looking for NODEs solutions of the following form:

$$u\left(\xi \right)=\sum _{i=1}^n{\mathcal{P}}_i\left(G\right){\left(G^{\prime }\right)}^i.$$

(3)

Here ${\mathcal{P}}_i\left(G\right)$ are n functionals depending on $G\left(\xi \right)$ that have to be determinated. It is not always possible to find the most general form of ${\mathcal{P}}_i\left(G\right)$, but we can look to rational solutions of the following form:

$${\mathcal{P}}_i\left(G\right)=\frac{{\mathcal{N}}_i\left(G\right)}{{\mathcal{D}}_i\left(G\right)}.$$

(4)

This particular choice allows us to see that the functional expansion method includes other well-known choices. For example, the generalized and improved $G^{\prime }/G$ method [55,56] asks for

$${\mathcal{P}}_i\left(G\right)=\frac{{\alpha }_i}{G^i}\equiv {\alpha }_i{G}^{-i};{\alpha }_i=const.,i=\{0,\dots n\}.$$

(5)

The approach from [57] corresponds to ${\mathcal{P}}_i=a_i{G}^{-i}+b_{i+1}{G}^{-i+1}$, while the $(v/w)$ method from [58] is recovered for ${\mathcal{P}}_i\sim {(1/w)}^i$ and $G^{\prime }=v$.

2.2. An Auxiliary Equation of the Reaction–Diffusion Type

The specific form of the functionals ${\mathcal{P}}_i\left(G\right)$ from (4) has to be correlated with the choice of the auxiliary equation. The more general this equation is, accepting wider classes of solutions, the greater the complexity of the physical phenomena that (2) will be able to describe. As our study is focused on the vcBKdV Equation (1), which incorporates terms of dispersion, dissipation and nonlinearity, it is quite natural to look for auxiliary equations with potential fr describing such phenomena. From this perspective, a very good candidate for an auxiliary equation could be a convection–reaction–diffusion equation of the following type:

$$G_t=\nabla (D\nabla G)-\nabla \left(vG\right)-Q.$$

(6)

Here $G(x,y,t)$ represents the variable of interest and it is known as a solution of (6); $D=D\left(G\right)$ is the diffusion coefficient and $v(x,y,t)$ is the field of velocities, while $Q=Q\left(G\right)$ describes the reaction phenomena. Equation (6) is an example of a second-order NPDE which can also be transformed into a NODE whose most general form is

$$G^{″}=\mathcal{M}\left(G\right){\left(G^{\prime }\right)}^2+\mathcal{R}\left(G\right)G^{\prime }+\mathcal{Q}\left(G\right).$$

It is interesting to note that, for $\mathcal{M}\left(G\right)=2G+M,M=const$, $\mathcal{R}\left(G\right)=R=const$ and $\mathcal{Q}\left(G\right)=Q=const$, the previous equation becomes

$$G^2{G}^{″}-(2G+M){\left(G^{\prime }\right)}^2-R{G}^2{G}^{\prime }-Q{G}^4=0,$$

(7)

which offers the advantage of being written as a generalized Riccati equation involving three parameters $M,$ $R,$ Q as follows:

$${\left(\frac{G^{\prime }}{G^2}\right)}^{\prime }=M{\left(\frac{G^{\prime }}{G^2}\right)}^2+R\left(\frac{G^{\prime }}{G^2}\right)+Q.$$

(8)

The solutions of (8) can be effectivelly written down, so (7) has known solutions and can be used as an auxiliary equation. The solutions of this reaction–diffusion equation will be used in our study to solve the vcBKdV Equation (1). More precisely, the following solutions of (8) will be used [59]:

Case 1: If $QM$ $>0,$ $R=0$:

$$\frac{G^{\prime }\left(\xi \right)}{{\left[G\left(\xi \right)\right]}^2}=\frac{\sqrt{MQ}\left[Pcos\left(\sqrt{MQ}\xi \right)+Nsin\left(\sqrt{MQ}\xi \right)\right]}{M\left[Ncos\left(\sqrt{MQ}\xi \right)-Psin\left(\sqrt{MQ}\xi \right)\right]};$$

(9)

Case 2: If $QM$ $<0,$$R=0$:

$$\frac{G^{\prime }\left(\xi \right)}{{\left[G\left(\xi \right)\right]}^2}=\frac{\sqrt{-MQ}\left[Pcosh\left(2\sqrt{-MQ}\xi \right)+Psinh\left(2\sqrt{-MQ}\xi \right)+C\right]}{M\left[Pcosh\left(2\sqrt{-MQ}\xi \right)+Psinh\left(2\sqrt{-MQ}\xi \right)-C\right]};$$

(10)

Case 3: When $R^2-4MQ$ ≥ $0,$ $R\ne 0,$ we obtain:

$$\frac{G^{\prime }\left(\xi \right)}{{\left[G\left(\xi \right)\right]}^2}=-\frac{R}{2M}-\frac{\sqrt{R^2-4MQ}\left[Pcosh\left(\frac{\sqrt{R^2-4MQ}}{2}\xi \right)+Nsinh\left(\frac{\sqrt{R^2-4MQ}}{2}\xi \right)\right]}{2M\left[Psinh\left(\frac{\sqrt{R^2-4MQ}}{2}\xi \right)+Ncosh\left(\frac{\sqrt{R^2-4MQ}}{2}\xi \right)\right]};$$

(11)

Case 4: When $R^2-4MQ$ $<0,$ $R\ne 0$:

$$\frac{G^{\prime }\left(\xi \right)}{{\left[G\left(\xi \right)\right]}^2}=-\frac{R}{2M}-\frac{\sqrt{4MQ-R^2}\left[Pcos\left(\frac{\sqrt{4MQ-R^2}}{2}\xi \right)+Nsin\left(\frac{\sqrt{4MQ-R^2}}{2}\xi \right)\right]}{2M\left[Psin\left(\frac{\sqrt{4MQ-R^2}}{2}\xi \right)+Ncos\left(\frac{\sqrt{4MQ-R^2}}{2}\xi \right)\right]},$$

(12)

where P, N, C are arbitrary constants.

3. Multiple Wave Solutions of the $(2+1)$ vcBKdV Equation

3.1. The Functional Expansion of the $(2+1)$ vcBKdV Equation

Let us now apply the two steps of the algorithmic method described above in order to construct multiple wave solutions for the $(2+1)$ Equation (1). In the first step, we transform the NPDE into a NODE using a wave variable $\xi (t,$ $x,$ $y)$ which is not compulsorily linear in its variables. It is assumed to be of the form

$$\xi (x,y,t)=kx+p(t,y),$$

(13)

with k being an arbitrary constant. The main solutions of (1) that we will analyse below will correspond to a $p(y,t)$ also linear in y but with an arbitrary nonlinear velocity function, $V\left(t\right)$:

$$\xi (x,y,t)=kx+sy+V\left(t\right).$$

(14)

In the second step of the functional expansion method, we will look for solutions of (1) in the form (3), with the specific choice in (4):

$${\mathcal{N}}_i\left(G\right)={\alpha }_i;{\mathcal{D}}_i\left(G\right)=G^{2i};{\mathcal{P}}_i={\alpha }_i{G}^{-2i}.$$

(15)

As vcBKdV (1) has variable coefficients, we will, as a supplement, consider that ${\alpha }_i={\alpha }_i\left(t\right)$. This choice corresponds to the $\left(\frac{G^{\prime }}{G^2}\right)$-expansion method [60], and it supposes that the solutions of Equation (1) are polynomials of $\left(\frac{G^{\prime }}{G^2}\right)$ with variable coefficients:

$$u(t,x,y)=\sum _{i=1}^n{\alpha }_i\left(t\right){\left[\frac{G^{\prime }\left(\xi \right)}{G{\left(\xi \right)}^2}\right]}^i.$$

(16)

In fact, it is assumed that $G\left(\xi \right)$ satisfies the more complex reaction–diffusion Equation (7), which is equivalent to the Riccati Equation (8) and accepts periodic and hyperbolic solutions given by (9)–(12). The positive integer n will be determined by taking into consideration the homogeneous balance between the highest order derivatives and the nonlinear terms involved in (1). For the determined n, by substituting (16) together with (7) into (1), then by setting all the coefficients of ${\left(\frac{G^{\prime }}{G^2}\right)}^i,$ $i=0,1,2,\dots$ to zero, one obtains an over-determined system with respect to the differential functions {$a\left(t\right),b\left(t\right),c\left(t\right),f\left(t\right),g\left(t\right)$} from (1), respectively, {${\alpha }_i(t,$ $x,$ $y),$ $i=\overline{1,n}$} from (16). This can be solved considering the constants $M,$ $R,$ Q, as well as $k,s$ and the velocity function $V\left(t\right)$ from (14), as supplementary parameters.

3.2. Explicit vcBKdV Wave Solutions

By replacing the solutions obtained from the previous determining system through symbolic computation packages like Mathematica or Maple, together with the known ones of the auxiliary Equation (8) into (16), a rich variety of explicit wave solutions of the master Equation (1) can be derived.

According to the rules mentioned before, we use the series expansion (16) as a form of solution for the vcBKdV Equation (1), where $G\left(\xi \right)$ can be any expression (9)–(12). By imposing the balance between the most nonlinear term $c\left(t\right)u_x^2$ and the higher-order derivative $a\left(t\right)u_{4x}$, we obtain $n=2$ and, by renaming the coefficients from (16), the generic solution of the master Equation (1) is assumed as follows:

$$u(t,x,y)=A\left(t\right)+B\left(t\right)\frac{G^{\prime }\left(\xi \right)}{G{\left(\xi \right)}^2}+H\left(t\right){\left[\frac{G^{\prime }\left(\xi \right)}{G{\left(\xi \right)}^2}\right]}^2,$$

(17)

where $A\left(t\right),$ $B\left(t\right),$ $H\left(t\right)$ are differentiable functions. The wave variable $\xi (x,y,t)$ has the form (13).

Substituting (17) along with (8) into (1), then setting the coefficients of ${\left(\frac{G^{\prime }}{G^2}\right)}^i,i=\overline{0,6}$ to zero, we obtain a determining system with respect to the unknown functions $A\left(t\right),$ $B\left(t\right),$ $H\left(t\right),$ $a\left(t\right),$ $b\left(t\right),$ $c\left(t\right),$ $f\left(t\right),$ $g\left(t\right),$ $p(t,$ $y)$. By solving this with the help of Maple program and coming back to (17), we can generate a rich set of vcBKdV solutions. We highlight the following three classes of interesting solutions below:

Solution 1: A family of $six$-parameter solutions can be obtained considering arbitrary constants $k,$ $s,$ $M,$ $Q,$ R and arbitrary function $V\left(t\right)$. We obtain

$$\begin{array}{c}a\left(t\right)=b\left(t\right)=c\left(t\right)=f\left(t\right)=g\left(t\right)=t,p(t,y)=sy+V\left(t\right),H\left(t\right)=\frac{60k^2{M}^2}{5},\\ A\left(t\right)=\frac{-k(30kR-1)\frac{dV\left(t\right)}{dt}-t\{(30kR+1)s^2+k^4[30Rk(8QM+R^2)+12(MQ+3R^2)]\}}{k^2(1+30kR)t},\\ B\left(t\right)=-\frac{12}{5}kM(1+5kR).\end{array}$$

(18)

Under conditions (18), the master vcBKdV Equation (1) admits the following family of $six$-parameter solutions:

$$\begin{array}{c}u(t,x,y)=\frac{-k(30kR-1)\frac{dV\left(t\right)}{dt}-t\{(30kR+1)s^2+k^4[30Rk(8QM+R^2)+12(MQ+3R^2)]\}}{k^2(1+30kR)t}\\ -\frac{12}{5}kM(1+5kR)\frac{G^{\prime }\left(\xi \right)}{G{\left(\xi \right)}^2}+\frac{60k^2{M}^2}{5}{\left[\frac{G^{\prime }\left(\xi \right)}{G{\left(\xi \right)}^2}\right]}^2,\xi (t,x,y)=kx+sy+V\left(t\right),\end{array}$$

(19)

available for whatever solution $\frac{G^{\prime }\left(\xi \right)}{G{\left(\xi \right)}^2}$ from (9)–(12).

Solution 2: For arbitrary constants $k,$ $s,$ $M,$ $Q,$ $R,$ $\rho$ and arbitrary functions $V\left(t\right),$ $A\left(t\right),$ $c\left(t\right),$ $f\left(t\right),$ $g\left(t\right)$, we can generate a family of 11-parameter wave solutions, obtaining

$$\begin{array}{c}p(t,y)=sy+V\left(t\right),H\left(t\right)=\frac{60k^2{M}^{2}a\left(t\right)}{3f\left(t\right)+2c\left(t\right)},\\ B\left(t\right)=\frac{300MR{k}^2}{I\left(t\right)+\rho },a\left(t\right)=\frac{3f\left(t\right)+2c\left(t\right)}{-I\left(t\right)-\rho },b\left(t\right)=\frac{6kR\left[17f\right(t)+8c(t\left)\right]}{I\left(t\right)+\rho },\\ I\left(t\right)={\displaystyle \int }-6R{k}^3[60MQf\left(t\right)+93R^{2}f\left(t\right)+82R^{2}c\left(t\right)-40MQc\left(t\right)]dt.\end{array}$$

(20)

The previous relations represent the general conditions under which the analyzed model (1) admits a family of 11-parameter wave solutions given by

$$\begin{array}{c}u(t,x,y)=A\left(t\right)+\frac{300MR{k}^2}{{\displaystyle \int }-6R{k}^3[60MQf\left(t\right)+93R^{2}f\left(t\right)+82R^{2}c\left(t\right)-40MQc\left(t\right)]dt+\rho }\frac{G^{\prime }\left(\xi \right)}{G{\left(\xi \right)}^2}\\ +\frac{60k^2{M}^{2}a\left(t\right)}{3f\left(t\right)+2c\left(t\right)}{\left[\frac{G^{\prime }\left(\xi \right)}{G{\left(\xi \right)}^2}\right]}^2,\xi (t,x,y)=kx+sy+V\left(t\right),\end{array}$$

(21)

where $\frac{G^{\prime }\left(\xi \right)}{G{\left(\xi \right)}^2}$ can take any of the forms (9)–(12).

Solution 3: For arbitrary constants $k,$ $s,$ $M,$ $Q,$ R and arbitrary functions $V\left(t\right),$ $B\left(t\right),$ $a\left(t\right),$ $c\left(t\right),$ $f\left(t\right),$ $g\left(t\right)$, we obtain

$$\begin{array}{c}p(t,y)=sy+V\left(t\right),H\left(t\right)=\frac{60k^2{M}^{2}a\left(t\right)}{3f\left(t\right)+2c\left(t\right)},A\left(t\right)=\frac{\alpha \left(t\right)}{\beta \left(t\right)},\\ b\left(t\right)=\frac{-60k^{2}MRa\left(t\right)[17f\left(t\right)+8c\left(t\right)]-51B\left(t\right){\left[f\left(t\right)\right]}^2-58B\left(t\right)f\left(t\right)c\left(t\right)-16B\left(t\right){\left[c\left(t\right)\right]}^2}{60kM\left[3f\right(t)+2c(t\left)\right]},\end{array}$$

(22)

where $\alpha \left(t\right)$ and $\beta \left(t\right)$ admit the following expressions:

$$\begin{array}{c}\alpha \left(t\right)=600M^2{R}^3{k}^6{\left[a\left(t\right)\right]}^2\{64f\left(t\right)+11c\left(t\right)\}+2190M{R}^2{k}^{4}a\left(t\right){\left[f\left(t\right)\right]}^{2}B\left(t\right)\\ -300M^2{k}^3\left\{3f\left(t\right)\frac{da\left(t\right)}{dt}+2c\left(t\right)\frac{da\left(t\right)}{dt}-3a\left(t\right)\frac{df\left(t\right)}{dt}\right\}+600M^2{k}^{3}a\left(t\right)\frac{dc\left(t\right)}{dt}\\ +10M{R}^2{k}^{4}a\left(t\right)c\left(t\right)B\left(t\right)\{197f\left(t\right)+34c\left(t\right)\}-9k^{2}R{\left[f\left(t\right)\right]}^3{\left[B\left(t\right)\right]}^2\\ +M{s}^{2}g\left(t\right)B\left(t\right)\{20{\left[c\left(t\right)\right]}^2+45{\left[f\left(t\right)\right]}^2+60f\left(t\right)c\left(t\right)\}\\ -1500M^2{k}^2{s}^{2}Rg\left(t\right)a\left(t\right)\{3f\left(t\right)+2c\left(t\right)\}-3k^{2}Rc\left(t\right){\left[B\left(t\right)\right]}^2{\left[f\left(t\right)\right]}^2\\ +MkB\left(t\right)\frac{dV\left(t\right)}{dt}\{20{\left[c\left(t\right)\right]}^2+45{\left[f\left(t\right)\right]}^2+60f\left(t\right)c\left(t\right)\}\\ +8k^{2}Rf\left(t\right){\left[c\left(t\right)\right]}^2{\left[B\left(t\right)\right]}^2-1500M^2{k}^{3}Ra\left(t\right)\frac{dV\left(t\right)}{dt}\{3f\left(t\right)+2c\left(t\right)\}\\ +100Q{M}^2{k}^{4}a\left(t\right)B\left(t\right)\{15{\left[f\left(t\right)\right]}^2+4f\left(t\right)c\left(t\right)-4{\left[c\left(t\right)\right]}^2\}\\ -6000RQ{M}^3{k}^6{\left[a\left(t\right)\right]}^2\{2c\left(t\right)+7f\left(t\right)\}4k^{2}R{\left[c\left(t\right)\right]}^3{\left[B\left(t\right)\right]}^2,\end{array}$$

(23)

$$\beta \left(t\right)=5k^{2}Mf\left(t\right)\{300k^{2}MRa\left(t\right)-B\left(t\right)[3f\left(t\right)+2c\left(t\right)]\}[3f\left(t\right)+2c\left(t\right)].$$

(24)

The previous relations between functional parameters ensure the conditions under which we can generate a more expanded family with 11-parameter wave solutions for vcBKdV equations. The structure of solutions is as follows:

$$\begin{array}{c}u(t,x,y)=\frac{\alpha \left(t\right)}{\beta \left(t\right)}+5k^{2}Mf\left(t\right)\{300k^{2}MRa\left(t\right)-B\left(t\right)[3f\left(t\right)+2c\left(t\right)]\}[3f\left(t\right)+2c\left(t\right)]\frac{G^{\prime }\left(\xi \right)}{G{\left(\xi \right)}^2}\\ +\frac{60k^2{M}^{2}a\left(t\right)}{3f\left(t\right)+2c\left(t\right)}{\left[\frac{G^{\prime }\left(\xi \right)}{G{\left(\xi \right)}^2}\right]}^2,\xi (t,x,y)=kx+sy+V\left(t\right),\end{array}$$

(25)

with $\alpha \left(t\right)$ and $\beta \left(t\right)$ given, respectively, by (23) and (24) and $\frac{G^{\prime }\left(\xi \right)}{G{\left(\xi \right)}^2}$ having taken expressions from (9)–(12).

4. Discussion of Some Specific Solutions

In this section, we will present the dynamical behaviours of some specific wave solutions of the $(2+1)$ vcBKdV Equation (1) belonging to each family of solutions obtained above.

First of all, let us mention that our results are more general than any others reported up until now in the literature. For example, the autonomous BKdV equation analysed in [50] is recovered if we consider that

$$\begin{array}{c}a\left(t\right)=c=const.,b\left(t\right)=b=const.,c\left(t\right)=f\left(t\right)=2m=const.,\\ g\left(t\right)=0,s=0,V\left(t\right)=-\omega t,\omega =const.\end{array}$$

(26)

For these choices, we obtain more general solutions than those from [50]. For example, our solution (21) leads to a solution of the following form:

$$\begin{array}{c}u(t,x,y)=A\left(t\right)+\frac{300MR{k}^2}{K_{1}t+K_2}\frac{G^{\prime }\left(\xi \right)}{G{\left(\xi \right)}^2}+K_3{\left[\frac{G^{\prime }\left(\xi \right)}{G{\left(\xi \right)}^2}\right]}^2,\\ M=R=K_1=K_2=K_3=const.,\xi (t,x)=kx-\omega t.\end{array}$$

For $\frac{G^{\prime }\left(\xi \right)}{G{\left(\xi \right)}^2}$ given by (10), it is a more general hyperbolic solution than in [50].

Our results (21) or (25) also generalize the finding from [53], where a $2D$ non-autonomo-us BKdV was studied. The equation from that paper is obtained as a particular case of (1) for

$$a\left(t\right)=-\gamma \sigma \left(t\right),\gamma =const.,b\left(t\right)=1,c\left(t\right)=f\left(t\right)=2\lambda =const.;g\left(t\right)=0.$$

Considering for example (21), the more general non-autonomous BKdV solutions admit the following expressions:

$$\begin{array}{c}u(t,x,y)=A\left(t\right)+\frac{300MR{k}^2}{N_{1}t+N_2}\frac{G^{\prime }\left(\xi \right)}{G{\left(\xi \right)}^2}-\frac{60k^2{M}^2\gamma \sigma \left(t\right)}{10\lambda }{\left[\frac{G^{\prime }\left(\xi \right)}{G{\left(\xi \right)}^2}\right]}^2,\forall A\left(t\right),\forall \sigma \left(t\right),\\ N_1=N_2=const.,\xi (t,x)=kx+V\left(t\right),\forall V\left(t\right),\end{array}$$

with $\frac{G^{\prime }\left(\xi \right)}{G{\left(\xi \right)}^2}$ given by (12).

Let us also point out the physical significance of some particular solutions that belong to the three families reported previously.

Figure 1 shows the periodic soliton wave solution $u(t,$ $2,$ $y)$ from (19), by choosing the arbitrary constants and arbitrary functions as

$$\begin{array}{c}k=s=R=2,Q=-5M=P=-\frac{N}{8}=\frac{1}{2},\\ a\left(t\right)=b\left(t\right)=c\left(t\right)=f\left(t\right)=g\left(t\right)=t,V\left(t\right)=5{\left[sin\left(t\right)\right]}^{2}cos\left(t\right),\end{array}$$

(27)

and by introducing the solitonic solution (11) of the auxiliary Riccati equation. As we see in the figure, we obtain a lattice which combines W- and M-shaped soliton waves. The conditions (27) and the constraints (18) are useful in easily determining the remaining involved functions, namely the wave variable $\xi (t,$ $x,$ $y)=kx+sy+V(t)$ and the variable coefficient functions $A\left(t\right),$ $B\left(t\right),H\left(t\right).$ It is important to note that, despite using a hyperbolic solution of the auxiliary equation, we obtained a periodic solution of Equation (1). The equations with variable coefficients do not inherit the solution type of the auxiliary equation to which they refer.

Figure 2 depicts the wave solution $u(t,$ $2,$ $y)$ from (21) with the following selection of parameters:

$$\begin{array}{c}s=\rho =R=4k=4M=4Q=4P=-4N=2,\\ c\left(t\right)=f\left(t\right)=t^{2}sinh\left(t\right),A\left(t\right)=V\left(t\right)=tsin\left(t\right),\forall g\left(t\right),\end{array}$$

(28)

and using, again, the Riccati hyperbolic solution (11). The wave profile appears when $R^2-4MQ$ $>0.$ The wave variable $\xi (t,x,y)$ and the coefficient functions $B\left(t\right),H\left(t\right),$ $a\left(t\right),b\left(t\right)$ are now given by introducing the choices (28) into the expressions (20).

Figure 3 shows the corresponding periodic waves by choosing the arbitrary constants and arbitrary functions as follows:

$$\begin{array}{c}N=Q=2M=-P=2k=2s=2\rho =4,R=3,\\ c\left(t\right)=t{\left[sin\left(t\right)\right]}^2,f\left(t\right)=t{\left[cos\left(t\right)\right]}^2,A\left(t\right)={\left[sin\left(t\right)\right]}^2,V\left(t\right)={\left[cos\left(t\right)\right]}^2,\forall g\left(t\right).\end{array}$$

(29)

Because, in this situation, $R^2-4MQ$ $<0,$ the Riccati trigonometric solution (12) has to be introduced into the multiple solution (21). The wave variable $\xi (t,x,y)$ and the coefficient functions $B\left(t\right),H\left(t\right),$ $a\left(t\right),b\left(t\right)$ admit the forms (20) which are now related to the chosen parameters (29).

It is important to now analyze how the solutions of Equation (1) depend on the choice of the auxiliary equation, in our case, on the parameters M, R and Q. For this, let us first consider the dynamical behaviours of the waves represented by the multiple solution (25) at $x=2$, for the same set of free parameters except the arbitrary constant Q. They are plotted in Figure 4 and Figure 5. These Figures exhibit the $3D$ profiles and the contour plots of the waves which have been obtained under the common conditions given by

$$\begin{array}{c}R=0,s=4k=M=-4P=4N=2,\\ a\left(t\right)=f\left(t\right)=g\left(t\right)={\left[cosh\left(t\right)\right]}^2,c\left(t\right)={\left[sinh\left(t\right)\right]}^2,\\ B\left(t\right)=sin\left(t\right),V\left(t\right)={\left[sin\left(t\right)\right]}^6,\end{array}$$

(30)

and, respectively, for distinct values $Q=0.001$ and $Q=0.5.$ In this case, the expression chosen for $\frac{G^{\prime }\left(\xi \right)}{G{\left(\xi \right)}^2}$ is of the trigonometric type (9).

In Figure 6, the dynamical characteristics of the wave solution $u(t,$ $2,$ $y)$ from (25) are depicted for the parameters chosen in (30), except the value of the Riccati parameter M which is now increased to $M=20.$ Because $QM$ $>0$, the Riccati solution (9) is preserved.

For Figure 4, Figure 5 and Figure 6, the wave function $\xi (t,$ $x,$ $y)=kx+sy+V(t)$ and the coefficient functions $b\left(t\right),$ $A\left(t\right),$ $H\left(t\right)$ can be easily determined using the expressions (22) and the appropriate conditions specified under each Figure.

Let us now study the second important issue raised by the functional expansion method: the dependence of the solutions on the form in which they are sought. We will consider, for this, three different expressions for the free function $B\left(t\right)$ from the vcBKdV solution (17). Figure 7 shows the $3D$ profiles of the wave solution from (25) at $x=4$, and imposes the following set of parameters:

$$\begin{array}{c}R=0,s=k=\frac{M}{2}=\frac{Q}{2}=\frac{P}{2}=\frac{N}{2}=1,a\left(t\right)=c\left(t\right)=e^{t^2},\\ a\left(t\right)=c\left(t\right)=e^{t^2}f\left(t\right)=g\left(t\right)=t^2,V\left(t\right)=\frac{t-sin\left(t^2\right)cos\left(t^2\right)}{2}.\end{array}$$

(31)

Under these conditions, we use the graphical representations of the harmonic Riccati solution (9), because $QM$ $>0$ and $R=0.$

5. Conclusions

This study focused on the generalized form of the Burgers–KdV equation with variable coefficients (1), considered in one temporal and two spatial dimensions. Three important aspects lend weight to our results: (i) the consideration in the initial equation of variable coefficients in time, as well as of variable propagation speeds; (ii) the use of a general solving method and the functional expansion approach that, as we already mentioned, includes other similar approaches based on auxiliary equations; and (iii) the choice of (7) as an auxiliary equation, a general reaction–diffusion equation which describes complex physical phenomena as dispersion, dissipation or convection and which is very specific for the Burgers–KdV equation. Practically, the form of the auxiliary equation was chosen to be equivalent to the generalized Riccati Equation (8), accepting (9)–(12) among its solutions. The procedure of generating wave solutions of Equation (1) supposed the use of the wave variable (14), with a nonlinear velocity function $V\left(t\right)$, and to substitute (17) and (8) into (1). Setting, then, the coefficients of ${\left(\frac{G^{\prime }}{G^2}\right)}^i,i=\overline{0,6}$ to zero, we obtained an overdetermined system of 7 ordinary differential equations with 14 unknown functions and parameters: the five coefficients, $a\left(t\right),b\left(t\right),$ $c\left(t\right),f\left(t\right)$ and $g\left(t\right)$, from the initial Equation (1); the three constant parameters, $M,R,Q$, from the auxiliary Equation (7); the three functions, $A\left(t\right),B\left(t\right)$ and $H\left(t\right)$, that appear in the sought form of solution (17); and the two constant parameters, k, s, and the velocity function $V\left(t\right)$ defining the wave variable (14).

By solving this determining system, we can practically generate a large number of non-autonomous solutions of the ($2+1$)-dimensional vcBKdV equation. The system was solved by imposing appropriate constraints and three specific families of solutions were pointed out: the family (19) of 6-parameter solutions and two families (21), respectively (25), of 11-parameter solutions. They were generated starting from the solutions (9)–(12), including periodic- and hyperbolic-type solutions. The wave variables $\xi (x,y,t)$ have a nonlinear velocity $V\left(t\right)$. To the best of our knowledge, such ($2+1$)-solutions have not yet been reported in the specialized literature. Our results are more general than the ones which have been obtained in the specific case of the autonomous ($2+1$)-BKdV equation [45] or in the ($1+1$)-dimensional non-autonomous cases [53,54].

Concerning the significance of the results, we investigated the influence of the two important factors that appear in the solving method: (i) the form of the auxiliary equation and (ii) the form in which the solutions are sought. The dependence of the wave propagations on the auxiliary equation was considered in (7) by taking, succesively, various values of the parameters, R, M and Q. The dependence on the form sought for solutions was studied by considering various $B\left(t\right)$ in (17). The reported results are very important for understanding the real phenomena that appear in hydrodynamics or in plasma physics. The applied technique can be useful for investigating other integrable or non-integrable nonlinear models in higher dimensions.