Exact Solutions of the (2+1)-Dimensional Generalized Burgers–Fisher Equation via Lie Symmetry and Generalized Bernoulli Method ^†

Abstract

This paper systematically studies the exact analytical solutions of the (2+1)-dimensional generalized Burgers–Fisher (gBF) equation. Using the Lie symmetry analysis method, the infinitesimal generators of the equation are derived. Through symmetry reduction, the original (2+1)-dimensional partial differential equation (PDE) is reduced to a (1+1)-dimensional equation, which is further transformed into an ordinary differential equation (ODE) via the traveling wave transformation. On this basis, a series of exact traveling wave solutions are successfully obtained by applying the generalized Bernoulli equation method, including hyperbolic tangent-type kink solitary wave solutions and hyperbolic cotangent-type singular soliton solutions. The study also conducts a visual analysis of the solution characteristics through three-dimensional graphs and contour plots. In particular, this paper discusses the case where the parameter n takes general values, filling the research gap in the existing literature.

1. Introduction

The Burgers equation and Fisher equation are nonlinear partial differential equations with broad applications across physics, chemistry, and biology [1,2,3,4]. These equations have been extensively studied due to their fundamental role in modeling complex systems. The generalized form of the Burgers–Fisher equation is given by

$$\begin{array}{cc}\hfill & u_t+h\left(u\right)u_x=u_{xx}+k\left(u\right),\hfill \end{array}$$

(1)

where the functions $h\left(u\right)$ and $k\left(u\right)$ allow for a range of applications in modeling various physical and biological phenomena, including nonlinear wave propagation and coupled reaction–diffusion processes. With different definitions of $h\left(u\right)$ and $k\left(u\right)$, this generalized equation can model diverse dynamic behaviors. When $h\left(u\right)=\alpha u^n$ and $k\left(u\right)=\beta u(1-u^n)$, Equation (1) becomes a most important (1+1)-dimensional generalized Burgers–Fisher equation [5], represented as

$$\begin{array}{cc}\hfill & u_t+\alpha u^n{u}_x-u_{xx}-\beta u(1-u^n)=0,\hfill \end{array}$$

(2)

where $\alpha$, $\beta$, and n are real constants. Equation (2) has broad applications in fluid mechanics, nonlinear optics, chemical physics, plasma physics, fluid physics, and traffic flow. In recent years, significant progress has been made in the study of high-dimensional Burgers-Fisher equations. For instance, the application of symmetry analysis in higher dimensions has provided new insights into the structure of solutions [6,7,8]. Additionally, the development of advanced computational techniques has enabled more accurate numerical simulations of these equations. Obtaining exact solutions for nonlinear PDEs is crucial in mathematical physics. Numerous techniques have been developed to find exact solutions, including the Hirota bilinear method [9], Darboux transformation [10], Lie symmetry [11], the Exp-function method [12], and the improved (G’/G)-expansion method [13]. The Exp-Function Method was first proposed by He and Wu in 2006 [14]. Its core advantage lies in efficiently obtaining various exact solutions (such as solitary wave solutions and periodic solutions) of nonlinear wave equations through a concise exponential function expansion form, and it is particularly suitable for complex nonlinear systems that are difficult to linearize via traditional transformations. In this study, we investigate a (2+1)-dimensional generalization of the Burgers–Fisher equation, the most general form of the Burgers–Fisher equation given by

$$\begin{array}{cc}\hfill & u_t+\alpha u^n(u_x+u_y)-u_{xx}-u_{yy}-\beta u(1-u^n)=0.\hfill \end{array}$$

(3)

In Section 2, we perform Lie symmetry analysis on Equation (3), reducing the equation. In Section 3, we apply the Generalized Bernoulli equation method to solve the ODE and get the exact solutions. Section 4 presents graphical representations and analysis. Finally, we conclude the study in Section 5.

2. Application of Lie Group Analysis

Symmetry represents the basic framework within nonlinear PDEs. It serves two key functions: it enables the discovery of invariant solutions, and it allows for the creation of transformations that map the solution set onto itself [15,16].

In this section, we present the infinitesimal symmetries and symmetry reductions of the gBF equation. Consider the one-parameter invariant transformation on $(x,y,t,u):$

$$\begin{array}{cc}\hfill & x^{*}=x+\epsilon \xi (x,y,t,u)+O\left({\epsilon }^2\right),\hfill \\ \hfill & y^{*}=y+\epsilon \eta (x,y,t,u)+O\left({\epsilon }^2\right),\hfill \\ \hfill & t^{*}=t+\epsilon \tau (x,y,t,u)+O\left({\epsilon }^2\right),\hfill \\ \hfill & u^{*}=u+\epsilon \varphi (x,y,t,u)+O\left({\epsilon }^2\right),\hfill \end{array}$$

(4)

where $\epsilon$ is the group parameter, $\xi ,\eta ,\tau$ and $\varphi$ represent the infinitesimals generators corresponding to $x,y,t$ and u, respectively. Each set of infinitesimals defines an infinitesimal symmetry, which leads to the following generator:

$$\begin{array}{cc}\hfill & V=\xi (x,y,t,u){\displaystyle \frac{\partial }{\partial x}}+\eta (x,y,t,u){\displaystyle \frac{\partial }{\partial y}}+\tau (x,y,t,u){\displaystyle \frac{\partial }{\partial t}}+\varphi (x,y,t,u){\displaystyle \frac{\partial }{\partial u}}.\hfill \end{array}$$

(5)

The second prolongation formula corresponding to this symmetry is given by

$$\begin{array}{cc}\hfill & {Pr}^{\left(2\right)}V=V+{\varphi }^x{\displaystyle \frac{\partial }{\partial u_x}}+{\varphi }^y{\displaystyle \frac{\partial }{\partial u_y}}+{\varphi }^t{\displaystyle \frac{\partial }{\partial u_t}}+{\varphi }^{xx}{\displaystyle \frac{\partial }{\partial u_{xx}}}+{\varphi }^{yy}{\displaystyle \frac{\partial }{\partial u_{yy}}},\hfill \end{array}$$

(6)

which ensures the invariance condition

$${Pr}^{\left(2\right)}V\left(u_t+\alpha u^n(u_x+u_y)-u_{xx}-u_{yy}-\beta u(1-u^n)\right){|}_{Equation \left(3\right)=0}=0.$$

(7)

Since there are no terms $u_{tt}$, $u_{tx}$, $u_{ty}$ and $u_{xy}$ in Equation (3), the coefficients of ${\varphi }^{tt}$, ${\varphi }^{tx}$, ${\varphi }^{ty}$, and ${\varphi }^{xy}$ are zero after expanding Equation (7).

The coefficients ${\varphi }^{xx},$ and ${\varphi }^{yy}$ are given by

$$\begin{array}{cc}\hfill {\varphi }^x& =D_x\left(\varphi \right)-u_t{D}_x\left(\tau \right)-u_x{D}_x\left(\xi \right)-u_y{D}_x\left(\eta \right),\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill {\varphi }^y& =D_y\left(\varphi \right)-u_t{D}_y\left(\tau \right)-u_x{D}_y\left(\xi \right)-u_y{D}_y\left(\eta \right),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\varphi }^t& =D_t\left(\varphi \right)-u_t{D}_t\left(\tau \right)-u_x{D}_t\left(\xi \right)-u_y{D}_t\left(\eta \right),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\varphi }^{xx}& =D_x\left({\varphi }^x\right)-u_{tx}{D}_x\left(\tau \right)-u_{xx}{D}_x\left(\xi \right)-u_{xy}{D}_x\left(\eta \right),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\varphi }^{yy}& =D_y\left({\varphi }^y\right)-u_{ty}{D}_y\left(\tau \right)-u_{xy}{D}_y\left(\xi \right)-u_{yy}{D}_y\left(\eta \right),\hfill \end{array}$$

(12)

where $D_t={\displaystyle \frac{\partial }{\partial t}}+u_t{\displaystyle \frac{\partial }{\partial u}}+u_{tt}{\displaystyle \frac{\partial }{\partial u_t}}+u_{tx}{\displaystyle \frac{\partial }{\partial u_x}}+u_{ty}{\displaystyle \frac{\partial }{\partial u_y}}+u_{ttt}{\displaystyle \frac{\partial }{\partial u_{tt}}}\dots ,D_x={\displaystyle \frac{\partial }{\partial x}}+u_x{\displaystyle \frac{\partial }{\partial u}}+u_{tx}{\displaystyle \frac{\partial }{\partial u_t}}+$$u_{xx}{\displaystyle \frac{\partial }{\partial u_x}}+u_{xy}{\displaystyle \frac{\partial }{\partial u_y}}+u_{ttx}{\displaystyle \frac{\partial }{\partial u_{tt}}}\dots ,D_y={\displaystyle \frac{\partial }{\partial y}}+u_y{\displaystyle \frac{\partial }{\partial u}}+u_{ty}{\displaystyle \frac{\partial }{\partial u_t}}+u_{xy}{\displaystyle \frac{\partial }{\partial u_x}}+u_{yy}{\displaystyle \frac{\partial }{\partial u_y}}+u_{tty}{\displaystyle \frac{\partial }{\partial u_{tt}}}\dots$ are total derivatives.

By solving Equation (7), the following symmetries are derived:

$$\begin{array}{cc}\hfill & V_1={\displaystyle \frac{\partial }{\partial x}},V_2={\displaystyle \frac{\partial }{\partial y}},V_3={\displaystyle \frac{\partial }{\partial t}}.\hfill \end{array}$$

(13)

Computing the Lie brackets:

$$[V_1,V_2]=0,[V_1,V_3]=0,[V_2,V_3]=0,$$

we find that these generate a 3-dimensional abelian Lie algebra. Since the algebra is abelian, the adjoint action is trivial: $\mathrm{Ad}(exp\left(\epsilon V_i\right))V_j=V_j$ for all $i,j$.

The general element is $V=a_1{V}_1+a_2{V}_2+a_3{V}_3$. Due to the trivial adjoint action, the one-dimensional optimal system consists of all linearly independent directions modulo scalar multiplication. After analyzing the projective space, we obtain the optimal system [17]:

$$\{V_1,V_2,V_3,\alpha V_1+V_2,\alpha V_1+\beta V_2+V_3\},\alpha ,\beta ,\gamma \in \mathbb{R}.$$

(14)

For simplicity, in the case where parameters are set to 1, we have the representative set:

$$\{V_1,V_2,V_3,V_1+V_2,V_1+V_2+V_3\}.$$

The gBF equation can be reduced as follows:

• Reduction by $V_1$.

The characteristic equations corresponding to $V_1$ are: ${\displaystyle \frac{dx}{1}}={\displaystyle \frac{dy}{0}}={\displaystyle \frac{dt}{0}}={\displaystyle \frac{du}{0}},$ which leads to the following relations:

$$\begin{array}{cc}\hfill & \xi =y,\eta =t,u=U(\xi ,\eta ).\hfill \end{array}$$

(15)

Substituting into Equation (3) results in the simplified form

$$\begin{array}{cc}\hfill & U_{\eta }+\alpha U^n{U}_{\xi }-U_{\xi \xi }-\beta U(1-U^n)=0.\hfill \end{array}$$

(16)

• Reduction by $V_2$.

The characteristic equations corresponding to $V_2$ are: ${\displaystyle \frac{dx}{0}}={\displaystyle \frac{dy}{1}}={\displaystyle \frac{dt}{0}}={\displaystyle \frac{du}{0}}$, leading to

$$\begin{array}{cc}\hfill & \xi =x,\eta =t,u=U(\xi ,\eta ).\hfill \end{array}$$

(17)

Substituting into the Equation (3) gives the simplified form

$$\begin{array}{cc}\hfill & U_{\eta }+\alpha U^n{U}_{\xi }-U_{\xi \xi }-\beta U(1-U^n)=0.\hfill \end{array}$$

(18)

• Reduction by $V_3$.

The characteristic equation corresponding to $V_3$ are: ${\displaystyle \frac{dx}{0}}={\displaystyle \frac{dy}{0}}={\displaystyle \frac{dt}{1}}={\displaystyle \frac{du}{0}}$, leading to

$$\begin{array}{cc}\hfill & \xi =x,\eta =y,u=U(\xi ,\eta ).\hfill \end{array}$$

(19)

Substituting these into the Equation (3) gives the simplified form

$$\begin{array}{cc}\hfill & \alpha U^n(U_{\xi }+U_{\eta })-U_{\xi \xi }-U_{\eta \eta }-\beta U(1-U^n)=0.\hfill \end{array}$$

(20)

• Reduction by $V_1+V_2$.

The characteristic equations corresponding to $V_1+V_2$ are: ${\displaystyle \frac{dx}{1}}={\displaystyle \frac{dy}{1}}={\displaystyle \frac{dt}{0}}={\displaystyle \frac{du}{0}},$ which leads to the following relations:

$$\begin{array}{cc}\hfill & \xi =x-y,\eta =t,u=U(\xi ,\eta ).\hfill \end{array}$$

(21)

Substituting into Equation (3) results in the simplified form

$$\begin{array}{cc}\hfill & U_{\eta }-2U_{\xi \xi }-\beta U(1-U^{\eta })=0.\hfill \end{array}$$

(22)

• Reduction by $V_1+V_2+V_3$.

The characteristic equation corresponding $V_1+V_2+V_3$ are: ${\displaystyle \frac{dx}{1}}={\displaystyle \frac{dy}{1}}={\displaystyle \frac{dt}{1}}={\displaystyle \frac{du}{0}}$, leading to

$$\begin{array}{cc}\hfill & \xi =x-y,\eta =x-t,u=U(\xi ,\eta ).\hfill \end{array}$$

(23)

Substituting these into the Equation (3) results in

$$\begin{array}{cc}\hfill & 2U_{\xi \xi }+2U_{\xi \eta }+U_{\eta \eta }-(\alpha U^n-1)U_{\eta }+\beta U(1-U^n)=0.\hfill \end{array}$$

(24)

3. Generalized Bernoulli Equation Method

This section applies the generalized Bernoulli equation method [18] to construct the exact solutions of the Equation (3).

3.1. Overview

The Bernoulli equation expression is as follows:

$$F^{\prime }\left(\zeta \right)=b_{1}F\left(\zeta \right)+b_2{F}^m\left(\zeta \right),$$

(25)

where $b_1$ and $b_2$ are constants. Through transformation: $F\left(\zeta \right)=G^r\left(\zeta \right)$, Equation (25) can be transformed into

$$G^{\prime }\left(\zeta \right)=B_{1}G\left(\zeta \right)+B_2{G}^{(m-1)r+1}\left(\zeta \right),$$

(26)

where $b_1=r{B}_1$, $b_2=r{B}_2$. Equation (26) is called the generalized Bernoulli equation, and its solution is

$$F\left(\zeta \right)=\left\{\begin{array}{c}\hfill \begin{array}{cc}\hfill & {[-{\displaystyle \frac{B_1}{2B_2}}(1+\tan \mathrm{h}\left({\displaystyle \frac{r(m-1)B_1}{2}}\zeta \right))]}^{{\displaystyle \frac{1}{r(m-1)}}},B_1{B}_2\ne 0,\hfill \\ \hfill & {\left[{\displaystyle \frac{1}{r(1-m)B_2\zeta }}\right]}^{{\displaystyle \frac{1}{r(m-1)}}},B_1=0,B_2\ne 0,\hfill \\ \hfill & {[-{\displaystyle \frac{B_1}{2B_2}}(1+\cot \mathrm{h}\left({\displaystyle \frac{r(m-1)B_1}{2}}\zeta \right))]}^{{\displaystyle \frac{1}{r(m-1)}}},B_1{B}_2\ne 0.\hfill \end{array}\end{array}\right.$$

(27)

When using the generalized Bernoulli equation method to solve nonlinear wave equations with positive power nonlinear terms, we can assume that the equation has the following form of solution:

$$u\left(\zeta \right)=\lambda G^p\left(\zeta \right),$$

(28)

where $\lambda \ne 0$ is an undetermined constant, and $G\left(\zeta \right)$ satisfies Equation (26).

3.2. Method Application

Perform the traveling wave transformation on Equation (3).

$$\begin{array}{cc}\hfill & u(x,y,t)=U\left(\xi \right),\xi =x+y-ct.\hfill \end{array}$$

(29)

Substituting into the generalized Burgers–Fisher equation gives:

$$\begin{array}{cc}\hfill & -c{U}^{\prime }+2\alpha U^n{U}^{\prime }-2U^{″}-\beta U(1-U^n)=0.\hfill \end{array}$$

(30)

Let $n=r$ and assume $O\left(u\right)=p$, $O\left(G^{\prime }\right)=(m-1)r+1$. By balancing the $U^n{U}^{\prime }$ and $U^{″}$ terms in Equation (30), we obtain $p=m-1$. Therefore, if $m=3$ then $p=2$. Thus, Equation (30) can be assumed to have the following form of solution

$$u\left(\xi \right)=\lambda G^2\left(\xi \right).$$

(31)

Substitute Equation (31) with Equation (26) at m = 3 into Equation (30), and make the coefficients of $G^2$, $G^{2+2r}$, and $G^{2+4r}$ equal to zero, solving the algebraic equation system, and replacing r with n, obtaining

$$B_2=B_2,B_1=-{\displaystyle \frac{\alpha }{2(n+1)}},\lambda ={\left({\displaystyle \frac{2B_{2}n+2B_2}{\alpha }}\right)}^{{\displaystyle \frac{1}{n}}},c={\displaystyle \frac{2{\alpha }^2+\beta +\beta n^2+2\beta n}{\alpha (n+1)}}.$$

(32)

By substituting Equations (27) and (32) into Equation (31), we obtain the following solitary wave solution and singular solitary wave solution for Equation (3)

$$u_1={\left({\displaystyle \frac{1}{2}}\left(1-tanh\left({\displaystyle \frac{\alpha n\xi }{2n+2}}\right)\right)\right)}^{{\displaystyle \frac{1}{n}}},$$

(33)

$$u_2={\left({\displaystyle \frac{1}{2}}\left(1-coth\left({\displaystyle \frac{\alpha n\xi }{2n+2}}\right)\right)\right)}^{{\displaystyle \frac{1}{n}}},$$

(34)

$$\xi =-{\displaystyle \frac{t\left(2{\alpha }^2+\beta +\beta n^2+2\beta n\right)}{\alpha (n+1)}}+x+y.$$

(35)

4. Graphical Analysis of Solutions

In this section, we present 3D and contour graphical illustrations of the solutions to the (2+1)-dimensional gBF equation. These illustrations are based on specific values assigned to the free parameters. Traveling wave solutions are classified into localized, singular, and periodic types. Singular traveling wave solutions characterize the behavior of nonlinear systems under extreme conditions and correspond to phenomena such as cotangent-type and tangent-type singular solutions.

Figure 1 represents solutions (33), where $\left(a\right)$, $\left(b\right)$, and $\left(c\right)$ show the kink solitary wave solutions. Among them, $\left(a\right)$ is the x-t plot, $\left(b\right)$ is the y-t plot, and $\left(c\right)$ is the x-y plot. The parameters for $\left(a\right)$ are: $\alpha =10$, $\beta =1$, $n=2$, $y=1$ with bounds $-1\le x\le 1$, $-1\le t\le 1$. For $\left(b\right)$: $\alpha =10$, $\beta =1$, $n=2$, $x=1$ with bounds $-1\le y\le 1$, $-1\le t\le 1$. For $\left(c\right)$: $\alpha =10$, $\beta =1$, $n=2$, $t=0$ with bounds $-2\le x\le 2$, $-2\le y\le 2$. Given the same parameters, the x- t and y-t plots are identical.

Figure 2 represents solutions (34), due to the domain and singularity of the coth function, we plot the graph of $|u_2|$. $\left(a\right)$ is the x-t plot, $\left(b\right)$ is the y-t plot, and $\left(c\right)$ is the x–y plot. The parameters for $\left(a\right)$ are $\alpha =3$, $\beta =-1$, $n=2$, $y=1$ with bounds $-5\le x\le 5$, $-5\le t\le 5$. For $\left(b\right)$: $\alpha =3$, $\beta =-1$, $n=2$, $x=1$ with bounds $-5\le y\le 5$, $-5\le t\le 5$. For $\left(c\right)$: $\alpha =3$, $\beta =-1$, $n=2$, $t=1$ with bounds $-5\le x\le 5$, $-5\le y\le 5$. Given the same parameters, the x–t and y–t plots show identical profiles.

5. Conclusions

This study systematically investigates the exact analytical solutions of the (2+1)-dimensional gBF equation through the Lie symmetry analysis and the generalized Bernoulli equation method. First, based on the Lie group theory, the infinitesimal generators of the equation are derived, and the original (2+1)-dimensional partial differential equation is reduced to a (1+1)-dimensional partial differential equation by using the symmetry reduction method, realizing the effective simplification of the problem. Furthermore, the equation is transformed into an ordinary differential equation through the traveling wave transformation, and a series of traveling wave solutions are successfully solved by using the generalized Bernoulli equation method, including the hyperbolic tangent-type kink solitary wave solution and the hyperbolic cotangent-type singular soliton solution.

The theoretical framework established in this study provides an effective approach for solving high-dimensional nonlinear partial differential equations, and is particularly suitable for complex mathematical models that simultaneously include convection terms, diffusion terms, and nonlinear source terms. Future research can be further extended to the exploration of other types of analytical solutions of this equation (such as periodic solutions and breather solutions), and numerical solution research can also be carried out by combining modern computing methods such as neural networks.