A New (3+1)-Dimensional Extension of the Kadomtsev–Petviashvili–Boussinesq-like Equation: Multiple-Soliton Solutions and Other Particular Solutions

Abstract

In this study, we focus on investigating a novel extended (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like (KPB-like) equation. Initially, we utilized the Lie symmetry method to determine the symmetry generator by considering the Lie invariance condition. Subsequently, by similar reduction, the equation becomes ordinary differential equations (ODEs). Exact analytical solutions were derived through the power series method, with a comprehensive proof of solution convergence. Employing the $(G^{\prime }/G^2)$-expansion method enabled the identification of trigonometric, exponential, and rational solutions of the equation. Furthermore, we established the auto-Bäcklund transformation of the equation. Multiple-soliton solutions were identified by utilizing Hirota’s bilinear method. The fundamental properties of these solutions were elucidated through graphical representations. Our results are of certain value to the interpretation of nonlinear problems.

1. Introduction

Nonlinear partial differential equations (NLPDEs) are frequently employed to characterize various intricate phenomena and system behaviors across natural and engineering domains [1,2,3]. These encompass phenomena like turbulence in fluid mechanics, meteorological weather forecasting, and phase transitions in material science. Typically characterized by nonlinear interaction and non-equilibrium behavior, such phenomena and system behaviors necessitate an apt mathematical framework for scientific modeling and analysis, a role effectively fulfilled by NLPDEs. The exact solutions of NLPDEs demonstrate the impact of nonlinear effects on the system’s dynamic behavior, crucial for understanding the formation and evolution of phenomena like chaos, vortices, soliton solutions, and more. Hence, accurately finding the exact solutions is vital for theoretical research and essential for advancing science and engineering [4,5,6].

Recent decades have seen mathematicians and scientists proposing various methods to solve exact solutions for rapidly developing NLPDEs, such as the Lie symmetry method (LSM) [7,8,9,10,11], Hirota’s bilinear method [12,13,14,15,16], auto-Bäcklund transformation [17,18,19,20], the $\left(G^{\prime }/G^2\right)$-expansion method [21,22,23,24], Darboux transformation [25,26,27], the tan–cot technique [28,29], the generalized Riccati equation expansion method [30], Painlevé’s analysis [31,32], the optimal homotopy asymptotic method (OHAM) [33], the residual power series method (RPSM) [34], the inverse scattering transform method [35], and so on [36,37,38]. Lie symmetry analysis method serves as a powerful mathematical tool for finding exact solutions of NLPDEs. It is suitable for a wide range of complex systems by revealing their symmetries, simplifying the equations, and verifying the correctness of the solutions. The $\left(G^{\prime }/G^2\right)$-expansion method is a simple, efficient, and visually clear approach with procedural steps suitable for computer implementation. This method provides a concise and straightforward procedure for constructing traveling wave solutions, particularly solitary wave solutions, for nonlinear evolution equations. Hirota’s bilinear method stands out as one of the most effective and elegant methods for solving PDEs. Beyond solving specific equations, it can also generate new integrable equations, effectively combining nonlinear problem analysis with algebraic methods. This method is particularly valuable in mathematical physics, theoretical physics, engineering science, and other fields, providing a powerful approach for studying integrable systems and soliton theory to unveil the deep structures of systems.

In recent years, many researchers have conducted comprehensive and systematic analyses and research on the KPB-like equation; it is used to describe waves with nonlinear characteristics, making nonlinear waves significantly important in various physical systems. Examples of these systems include shallow water waves, plasma waves, and optical pulses in optical fibers. The KPB-like equation

$$u_{xxxy}+3{\left(u_x{u}_y\right)}_x+u_{xt}+u_{yt}+u_{tt}-u_{zz}=0,$$

(1)

first proposed by V.E. Zakharov and A.B. Shabat, combines features of the generalized Boussinesq equation

$$u_{xxxy}+3{\left(u_x{u}_y\right)}_x+u_{tt}-u_{zz}=0$$

(2)

and the generalized Kadomtsev–Petviashvili equation

$$u_{xxxy}+3{\left(u_x{u}_y\right)}_x+u_{xt}+u_{yt}-u_{zz}=0.$$

(3)

The KPB-like equation is well suited for describing nonlinear phenomena in water wave dynamics due to the integration of wave theory from the generalized Boussinesq equation with characteristics of the generalized Kadomtsev–Petviashvili equation.

By constructing the direct bilinear Bäcklund transformation, Yu and Sun obtained the exponential and rational traveling wave solutions of Equation (1) [39]. In addition, Sun et al. [40] investigated lump solutions in two special cases using the generalized bilinear method and derived the analytical and localized conditions for lump solitary waves. By employing Hirota’s method and symbolic computation, Sudao Bilige et al. [41] derived the lump, interaction, and breather-wave solutions within specific constraints. Then, they utilized the bilinear neural network method to create three neural network models for deriving periodic interaction solutions, respiratory solutions, rogue wave solutions, as well as bright- and dark-soliton solutions of Equation (1) [42]. By utilizing the multiple exp-function method and the multiple rogue wave solution method, Jalil Manafian was able to determine the multiple-soliton solutions of Equation (1), ranging from first-order to fourth-order wave solutions. Furthermore, the N-soliton wave solutions of Equation (1) were established through the application of the linear superposition principle [43].

In this paper, inspired by the extended equation in [44,45], we intend to study the solutions of the following new extended (3+1)-dimensional KPB-like equation:

$$u_{xxxy}+3{\left(u_x{u}_y\right)}_x+u_{tx}+u_{ty}+u_{tt}+\alpha u_{xx}+\beta u_{yy}-\gamma u_{zz}=0,$$

(4)

where $\alpha ,\beta$, and $\gamma$ are arbitrary constants. When $\alpha =\beta =0,\gamma =1$, Equation (4) is transformed into Equation (1). The new extended KPB-like equation proposed in this paper can reveal important physical properties and significance in nonlinear optics more comprehensively. This equation is also capable of describing the propagation and interaction of optical solitons in light, which is crucial for enhancing the stability and efficiency of optical communication systems. The study of this new equation not only has a profound impact on mathematical physics but also extends to various application fields. It provides essential theoretical support and numerical tools for solving practical problems and optimizing engineering design. In this paper, we mainly study the exact solutions of the equation and analyze the physical structure and characteristics of the exact solutions.

The structure of this paper is organized as follows: The integrability of the equation is established, and two nonlinear fourth-order ODEs are derived through similarity reductions in Section 2. In Section 3, the power series solution is determined via the power series method, with a proof of convergence provided. In Section 4, three different kinds of solutions are identified using the ($G^{\prime }/G^2$)-expansion method. Section 5 focuses on the auto-Bäcklund transformation of the equation achieved through Hirota’s bilinear method, leading to the discovery of single-soliton solutions, two-soliton solutions, and three-soliton solutions. The graphical representations of the solutions are presented in Section 6, followed by a comprehensive summary of the paper in the last section.

2. Painlevé Analysis and Lie Symmetry Analysis of the KPB-like Equation

2.1. Painlevé Analysis

Painlevé analysis is an effective method to prove whether the equation is integrable or not, and it is convenient to understand the characteristics of the equation [31]. Assume the solution of Equation (4) is explored as

$$u(x,y,z,t)={\displaystyle \sum _{k=0}^{\infty }}{u}_k(x,y,z,t)\varphi {(x,y,z,t)}^{k-j},$$

(5)

which is a Laurent series with respect to a singular manifold $\varphi (x,y,z,t)$. First, to obtain the leading term as well as the leading coefficient, substitute

$$u(x,y,z,t)=u_0(x,y,z,t){\varphi }^{-j}$$

(6)

into Equation (4), through the homogeneous balance method, we obtain $j=1,u_0(x,y,z,t)=2{\varphi }_x$. In order to obtain the resonant points, insert

$$u(x,y,z,t)=u_0{\varphi }^{-1}+u_k{\varphi }^{k-1}$$

(7)

and (5) into Equation (4), following the Weiss–Tabor–Carnevale (WTC) analysis, and by balancing the most dominant terms, we obtain the resonance points −1, 1, 4, and 6 due to the fourth-order of the linear structure of the equation. We thus obtain

$$u(x,y,z,t)=u_0{\varphi }^{-1}+u_1+u_2\varphi +u_3{\varphi }^2+u_4{\varphi }^3+u_5{\varphi }^4+u_6{\varphi }^5.$$

(8)

Substitute (8) into Equation (4) so that the coefficients of the same power of $\varphi$ equals zero. As a result, we obtain explicit expressions for $u_2,u_3,\mathrm{and}{u}_5$. However, we find that $u_1,u_4,u_6$ turn out to be arbitrary functions for all real-value nonzero parameters. Consequently, the new extended KPB-like equation is Painlevé-integrable.

2.2. Lie Symmetry Analysis

We achieve similarity in an extended (3+1)-dimensional KPB-like equation through Lie symmetry analysis, utilizing the one-parameter Lie group of infinitesimal transformations:

$$\begin{array}{c}\tilde{x}=x+\varsigma {\xi }^1(x,y,z,t,u)+O\left({\varsigma }^2\right),\\ \tilde{y}=y+\varsigma {\xi }^2(x,y,z,t,u)+O\left({\varsigma }^2\right),\\ \tilde{z}=z+\varsigma {\xi }^3(x,y,z,t,u)+O\left({\varsigma }^2\right),\\ \tilde{t}=t+\varsigma \tau (x,y,z,t,u)+O\left({\varsigma }^2\right),\\ \tilde{u}=u+\varsigma \eta (x,y,z,t,u)+O\left({\varsigma }^2\right),\end{array}$$

(9)

where ${\xi }^1,{\xi }^2,{\xi }^3,\tau ,$ and $\eta$ are the generators of Lie group transformations with a continuous group parameter $\varsigma$. The associated vector field is represented by

$$X={\xi }^1(x,y,z,t,u){\partial }_x+{\xi }^2(x,y,z,t,u){\partial }_y+{\xi }^3(x,y,z,t,u){\partial }_z+\tau (x,y,z,t,u){\partial }_t+\eta (x,y,z,t,u){\partial }_u.$$

(10)

The fourth prolongation of X for the Equation (4) is given by

$${\mathrm{Pr}}^{\left(4\right)}X\left(F\right){|}_{F=0}=0,$$

(11)

where $F=u_{xxxy}+3{\left(u_x{u}_y\right)}_x+u_{tx}+u_{ty}+u_{tt}+\alpha u_{xx}+\beta u_{yy}-\gamma u_{zz}.$ ${\mathrm{Pr}}^{\left(4\right)}X$ is defined as

$$\begin{array}{cc}& {\mathrm{Pr}}^{\left(4\right)}X=X+{\eta }^x{\displaystyle \frac{\partial }{\partial u_x}}+{\eta }^y{\displaystyle \frac{\partial }{\partial u_y}}+{\eta }^{xx}{\displaystyle \frac{\partial }{\partial u_{xx}}}+{\eta }^{xy}{\displaystyle \frac{\partial }{\partial u_{xy}}}+{\eta }^{tx}{\displaystyle \frac{\partial }{\partial u_{tx}}}+{\eta }^{ty}{\displaystyle \frac{\partial }{\partial u_{ty}}}\hfill \\ & +{\eta }^{tt}{\displaystyle \frac{\partial }{\partial u_{tt}}}+{\eta }^{yy}{\displaystyle \frac{\partial }{\partial u_{yy}}}+{\eta }^{zz}{\displaystyle \frac{\partial }{\partial u_{zz}}}+{\eta }^{xxxy}{\displaystyle \frac{\partial }{\partial u_{xxxy}}},\hfill \end{array}$$

(12)

and

$$\begin{array}{c}{\eta }^x=D_x(\eta -{\xi }^1{u}_x-{\xi }^2{u}_y-{\xi }^3{u}_z-\tau u_t)+{\xi }^1{u}_{xx}+{\xi }^2{u}_{xy}+{\xi }^3{u}_{xz}+\tau u_{xt},\\ {\eta }^y=D_y(\eta -{\xi }^1{u}_x-{\xi }^2{u}_y-{\xi }^3{u}_z-\tau u_t)+{\xi }^1{u}_{xy}+{\xi }^2{u}_{yy}+{\xi }^3{u}_{yz}+\tau u_{yt}\\ {\eta }^{xx}=D_x^2(\eta -{\xi }^1{u}_x-{\xi }^2{u}_y-{\xi }^3{u}_z-\tau u_t)+{\xi }^1{u}_{xxx}+{\xi }^2{u}_{xxy}+{\xi }^3{u}_{xxz}+\tau u_{xxt},\\ {\eta }^{xt}=D_x{D}_t(\eta -{\xi }^1{u}_x-{\xi }^2{u}_y-{\xi }^3{u}_z-\tau u_t)+{\xi }^1{u}_{xxt}+{\xi }^2{u}_{xyt}+{\xi }^3{u}_{xzt}+\tau u_{xtt},\\ {\eta }^{yt}=D_y{D}_t(\eta -{\xi }^1{u}_x-{\xi }^2{u}_y-{\xi }^3{u}_z-\tau u_t)+{\xi }^1{u}_{xyt}+{\xi }^2{u}_{yyt}+{\xi }^3{u}_{yzt}+\tau u_{ytt}\\ {\eta }^{tt}=D_t^2(\eta -{\xi }^1{u}_x-{\xi }^2{u}_y-{\xi }^3{u}_z-\tau u_t)+{\xi }^1{u}_{xtt}+{\xi }^2{u}_{ytt}+{\xi }^3{u}_{ztt}+\tau u_{ttt},\\ {\eta }^{yy}=D_y^2(\eta -{\xi }^1{u}_x-{\xi }^2{u}_y-{\xi }^3{u}_z-\tau u_t)+{\xi }^1{u}_{xyy}+{\xi }^2{u}_{yyy}+{\xi }^3{u}_{yyz}+\tau u_{yyt},\\ {\eta }^{zz}=D_z^2(\eta -{\xi }^1{u}_x-{\xi }^2{u}_y-{\xi }^3{u}_z-\tau u_t)+{\xi }^1{u}_{xzz}+{\xi }^2{u}_{yzz}+{\xi }^3{u}_{zzz}+\tau u_{zzt},\\ {\eta }^{xxxy}=D_x^3{D}_y(\eta -{\xi }^1{u}_x-{\xi }^2{u}_y-{\xi }^3{u}_z-\tau u_t)+{\xi }^1{u}_{xxxxy}+{\xi }^2{u}_{xxxyy}+{\xi }^3{u}_{xxxyz}+\tau u_{xxxyt},\end{array}$$

(13)

where $D_x,D_y,D_z,\mathrm{and}{D}_t$ are the total differentials dependent on $x,y,z$, and t, respectively. Their expression can be obtained by $D_i={\displaystyle \frac{\partial }{\partial x_i}}+u_i{\displaystyle \frac{\partial }{\partial u}}+u_{ij}{\displaystyle \frac{\partial }{\partial u_{ij}}}+\cdots ,i=1,2,3,4.$

After solving the equations above, setting the partial derivative coefficient of u to zero, and utilizing Maple 2024, we can determine the equations as follows:

$$\begin{array}{cc}\hfill & {\xi }^1={\displaystyle \frac{c_1}{3}}x+{\displaystyle \frac{c_1}{3}}t+{\displaystyle \frac{c_2}{2\gamma }}+c_6,\hfill \\ \hfill & {\xi }^2=c_{1}y+{\displaystyle \frac{c_2}{2\gamma }}z+c_5,\hfill \\ \hfill & {\xi }^3=c_{1}z+c_{2}t+c_3,\hfill \\ \hfill & \tau =c_{1}t+{\displaystyle \frac{c_2}{\gamma }}z+c_4,\hfill \\ \hfill & \eta ={\displaystyle \frac{x-3u+(1-4\alpha )y}{9}}{c}_1+f_1(t-{\displaystyle \frac{z}{\sqrt{\gamma }}})+f_2(t+{\displaystyle \frac{z}{\sqrt{\gamma }}}),\hfill \end{array}$$

(14)

where $c_1,c_2,c_3,c_4,c_5,\mathrm{and}{c}_6$ are arbitrary constants, and the vector fields of the symmetric group satisfying Equation (4) are

$$\begin{array}{cc}\hfill & X_1={\displaystyle \frac{\partial }{\partial x}},X_2={\displaystyle \frac{\partial }{\partial y}},X_3={\displaystyle \frac{\partial }{\partial z}},X_4={\displaystyle \frac{\partial }{\partial t}},X_5=z{\displaystyle \frac{\partial }{\partial x}}+z{\displaystyle \frac{\partial }{\partial y}}+2\gamma t{\displaystyle \frac{\partial }{\partial z}}+2z{\displaystyle \frac{\partial }{\partial t}},\hfill \\ \hfill & X_6=(3x+3t){\displaystyle \frac{\partial }{\partial x}}+9y{\displaystyle \frac{\partial }{\partial y}}+9z{\displaystyle \frac{\partial }{\partial z}}+9t{\displaystyle \frac{\partial }{\partial t}}+(x-3u-(4\alpha -1)y){\displaystyle \frac{\partial }{\partial u}},\hfill \\ \hfill & X_7=f_1(t-{\displaystyle \frac{z}{\sqrt{\gamma }}}){\displaystyle \frac{\partial }{\partial u}},X_8=f_2(t+{\displaystyle \frac{z}{\sqrt{\gamma }}}){\displaystyle \frac{\partial }{\partial u}}.\hfill \end{array}$$

(15)

By using the generators, we obtain the one-parameter transformation group of Equation (4) as

$$\begin{array}{cccc}\hfill & W_1:(x,y,z,t,u)\hfill & \hfill & \to (x+\varsigma ,y,z,t,u),\hfill \\ \hfill & W_2:(x,y,z,t,u)\hfill & \hfill & \to (x,y+\varsigma ,z,t,u),\hfill \\ \hfill & W_3:(x,y,z,t,u)\hfill & \hfill & \to (x,y,z+\varsigma ,t,u),\hfill \\ \hfill & W_4:(x,y,z,t,u)\hfill & \hfill & \to (x,y,z,t+\varsigma ,u),\hfill \\ \hfill & W_5:(x,y,z,t,u)\hfill & \hfill & \to (x+\varsigma z,y+\varsigma z,z+2\gamma \varsigma t,t+2\varsigma z,u),\hfill \\ \hfill & W_6:(x,y,z,t,u)\hfill & \hfill & \to (x+3\varsigma x+3\varsigma t,y+9\varsigma y,z+9\varsigma z,t+9\varsigma t,u+\varsigma x-(4\alpha -1)\varsigma y-3\varsigma u).\hfill \end{array}$$

(16)

According to the above one-parameter invariant groups $W_1,W_2,W_3,W_4,W_5,\mathrm{and}{W}_6$, we obtain the theory of Lie symmetry if $u=f(x,y,z,t)$ is a solution of the following equation:

$$\begin{array}{cc}\hfill & u^{\left(1\right)}=f(x-\varsigma ,y,z,t),\hfill \\ \hfill & u^{\left(2\right)}=f(x,y-\varsigma ,z,t),\hfill \\ \hfill & u^{\left(3\right)}=f(x,y,z-\varsigma ,t),\hfill \\ \hfill & u^{\left(4\right)}=f(x,y,z,t-\varsigma ),\hfill \\ \hfill & u^{\left(5\right)}=f(x-\varsigma z,y-\varsigma z,z-2\gamma \varsigma t,t-2\varsigma z),\hfill \\ \hfill & u^{\left(6\right)}=f(x-3\varsigma x-3\varsigma t,y-9\varsigma y,z-9\varsigma z,t-9\varsigma t).\hfill \end{array}$$

(17)

2.3. Similarity Reductions

In this section, the reduction equations are utilized to derive group invariant solutions for Equation (4). To obtain the reduction equations, invariant functions play a vital role by assisting in their identification. To start, the associated Lagrange’s system is solved in order to determine the invariant functions, which serve as the constant of integration for the characteristic equations below:

$$\begin{array}{cc}& {\displaystyle \frac{dx}{{\xi }^1(x,y,z,t,u)}}={\displaystyle \frac{dy}{{\xi }^2(x,y,z,t,u)}}={\displaystyle \frac{dz}{{\xi }^3(x,y,z,t,u)}}\hfill \\ & ={\displaystyle \frac{dt}{\tau (x,y,z,t,u)}}={\displaystyle \frac{du}{\eta (x,y,z,t,u)}}.\hfill \end{array}$$

(18)

Invariant functions obtained from characteristic equations play a crucial role in the creation of invariant solutions. These group invariant solutions exhibit invariance under the Lie symmetry group of a PDE. By solving a system of differential equations with one less independent variable than the original equation, one can derive solutions that remain invariant under a one-parameter Lie group of transformations.

Through the Lie symmetry equation, we choose two kinds of symmetries ${\mu }_1{X}_1+{\mu }_2{X}_2+{\mu }_3{X}_3+X_4,X_5$ as examples to illustrate the superiority and reliability of Lie symmetry method, where ${\mu }_1,{\mu }_2,\mathrm{and}{\mu }_3$ are arbitrary constants.

2.3.1. Symmetry ${\mu }_1{X}_1+{\mu }_2{X}_2+{\mu }_3{X}_3+X_4={\mu }_1{\displaystyle \frac{\partial }{\partial x}}+{\mu }_2{\displaystyle \frac{\partial }{\partial y}}+{\mu }_3{\displaystyle \frac{\partial }{\partial z}}+{\displaystyle \frac{\partial }{\partial t}}$

The Lagrange’s system of characteristic equation is given by

$${\displaystyle \frac{dx}{{\mu }_1}}={\displaystyle \frac{dy}{{\mu }_2}}={\displaystyle \frac{dz}{{\mu }_3}}={\displaystyle \frac{dt}{1}}={\displaystyle \frac{du}{0}}.$$

(19)

The invariant functions generated by the similarity reduction of Equation (4) are

$$u(x,y,z,t)=U(X,Y,Z),X=x-{\mu }_{1}t,Y=y-{\mu }_{2}t,Z=z-{\mu }_{3}t,$$

(20)

which reduces Equation (4) to

$$\begin{array}{cc}& U_{XXXY}+3{\left(U_X{U}_Y\right)}_X+({\mu }_1^2-{\mu }_1+\alpha )U_{XX}+(2{\mu }_1{\mu }_2-{\mu }_1-{\mu }_2)U_{XY}+(2{\mu }_1{\mu }_3-{\mu }_3)U_{XZ}\hfill \\ & +({\mu }_2^2-{\mu }_2+\beta )U_{YY}+(2{\mu }_2{\mu }_3-{\mu }_3)U_{YZ}+({\mu }_3^2-\gamma )U_{ZZ}=0.\hfill \end{array}$$

(21)

By applying the similarity transformation method to Equation (21), we discovered that it possesses three translation symmetries: $Z_1={\displaystyle \frac{\partial }{\partial X}},Z_2={\displaystyle \frac{\partial }{\partial Y}},Z_3={\displaystyle \frac{\partial }{\partial Z}}.$ Symmetry $s_1{Z}_1+s_2{Z}_2+Z_3$ produces the invariants

$$H(p,q)=U(X,Y,Z),p=X-s_{1}Z,q=Y-s_{2}Z$$

(22)

with $s_1,s_2$ as arbitrary constants. Substituting (22) into Equation (21), we obtain

$$H_{pppq}+3H_{pp}{H}_q+3H_{pq}{H}_p+A{H}_{pp}+B{H}_{pq}+C{H}_{qq}=0,$$

(23)

where

$$\begin{array}{cc}& A={\mu }_1^2-{\mu }_1+\alpha -2{\mu }_1{\mu }_3{s}_1+s_1{\mu }_3+{\mu }_3^2{s}_1^2-\gamma s_1^2,\hfill \\ & B=2{\mu }_1{\mu }_2-{\mu }_1-{\mu }_2-2{\mu }_1{\mu }_3{s}_2+s_2{\mu }_3-2{\mu }_2{\mu }_3{s}_1+s_1{\mu }_3+2{\mu }_3^2{s}_1{s}_2-2s_1{s}_2\gamma ,\hfill \\ & C={\mu }_2^2-{\mu }_2+\beta -2{\mu }_2{\mu }_3{s}_2+s_2{\mu }_3+s_2^2{\mu }_3^2-\gamma s_1^2.\hfill \end{array}$$

(24)

Equation (23)’s infinitesimal generators are

$$\begin{array}{cc}\hfill & {\xi }_p={\displaystyle \frac{a_{1}p}{3}}+a_3,\hfill \\ \hfill & {\xi }_q=a_{1}q+a_2,\hfill \\ \hfill & {\eta }_H=-{\displaystyle \frac{a_{1}H}{3}}-{\displaystyle \frac{4A{a}_{1}q}{9}}-{\displaystyle \frac{2B{a}_{1}p}{9}}+a_4,\hfill \end{array}$$

(25)

where ${\xi }_p,{\xi }_q,\mathrm{and}{\eta }_H$ denote infinitesimal generators with respect to the indicated variable; and $a_1,a_2,a_3$, and $a_4$ are arbitrary constants. If $a_1\ne 0,$ else parameters are zeros, and then we find a new similarity form of solution $R\left(w\right)$ and, hence, $H(p,q)$ can be rewritten as

$$H(p,q)={\displaystyle \frac{R\left(w\right)}{p}}-{\displaystyle \frac{B^2{p}^2}{12Aq}}-{\displaystyle \frac{Bp}{3}}-{\displaystyle \frac{Aq}{3}},$$

(26)

where $w={\displaystyle \frac{q}{p^3}}$ is a similarity variable.

Therefore, (26) reduces Equation (23) into the following ODE:

$$\begin{array}{cc}& (C-{\displaystyle \frac{15B^2}{4A}})R^{″}-3(40R^{\prime }+128w{R}^{″}+72w^2{R}^{″\prime }+9w^3{R}^{″″})\hfill \\ & +9(2R{R}^{\prime }+wR{R}^{″}+10w{R}^{\prime 2}+6w^2{R}^{\prime }{R}^{″})+{\displaystyle \frac{9B^2}{2A}}{w}^{-1}{R}^{\prime }-({\displaystyle \frac{B^4}{8A^2}}+{\displaystyle \frac{B^{2}C}{6A}})w^{-3}=0.\hfill \end{array}$$

(27)

2.3.2. Symmetry $X_5=z{\displaystyle \frac{\partial }{\partial x}}+z{\displaystyle \frac{\partial }{\partial y}}+2\gamma t{\displaystyle \frac{\partial }{\partial z}}+2z{\displaystyle \frac{\partial }{\partial t}}$

The associated Lagrange’s characteristic equations can be obtained as follows:

$${\displaystyle \frac{dx}{z}}={\displaystyle \frac{dy}{z}}={\displaystyle \frac{dz}{2\gamma t}}={\displaystyle \frac{dt}{2z}}={\displaystyle \frac{du}{0}}.$$

(28)

Equation (4) can be reduced into the following similarity form with new similarity variables:

$$u(x,y,z,t)=U(X,Y,Z),X=x-{\displaystyle \frac{t}{2}},Y=y-{\displaystyle \frac{t}{2}},Z=z^2-\gamma t^2.$$

(29)

We obtain the following PDE:

$$\begin{array}{cc}& U_{XXXY}+3U_X{U}_{XY}+3U_{XX}{U}_Y+(\alpha -{\displaystyle \frac{1}{4}})U_{XX}-{\displaystyle \frac{1}{2}}{U}_{XY}+(\beta -{\displaystyle \frac{1}{4}})U_{YY}\hfill \\ & -4\gamma U_Z-4\gamma Z{U}_{ZZ}=0.\hfill \end{array}$$

(30)

The symmetries of Equation (30) are

$$\begin{array}{cc}& {\overline{X}}_1={\displaystyle \frac{\partial }{\partial X}},{\overline{X}}_2={\displaystyle \frac{\partial }{\partial Y}},{\overline{X}}_3={\displaystyle \frac{\partial }{\partial U}},{\overline{X}}_4=(lnZ){\displaystyle \frac{\partial }{\partial U}},\hfill \\ & {\overline{X}}_5=3X{\displaystyle \frac{\partial }{\partial X}}+9Y{\displaystyle \frac{\partial }{\partial Y}}+18Z{\displaystyle \frac{\partial }{\partial Z}}+(X+(1-4\alpha )Y+3U){\displaystyle \frac{\partial }{\partial U}}.\hfill \end{array}$$

(31)

By utilizing the linear combination ${\overline{X}}_1-{\overline{X}}_2={\displaystyle \frac{\partial }{\partial X}}-{\displaystyle \frac{\partial }{\partial Y}}$, the invariant solution of Equation (30) is

$$U(X,Y,Z)=G(r,s),r=X+Y,s=Z.$$

(32)

Equation (30) has been reduced into the following PDE:

$$G_{rrrr}+6G_r{G}_{rr}+(\alpha +\beta -1)G_{rr}-4\gamma G_s-4\gamma s{G}_{ss}=0.$$

(33)

Generators of infinitesimal transformations for Equation (33) are

$${\xi }_r={\displaystyle \frac{b_{1}r}{4}}+b_2,{\xi }_s=b_{1}s,{\eta }_G=(lns)b_4+{\displaystyle \frac{(1-\alpha -\beta )r-3G}{12}}{b}_1+b_3$$

(34)

with $b_1,b_2,b_3,\mathrm{and}{b}_4$ as arbitrary constants. Hence, the Lagrange system of Equation (33) is

$$G(r,s)={\displaystyle \frac{R\left(w\right)}{r}}+{\displaystyle \frac{1-\alpha -\beta }{6}}r,w={\displaystyle \frac{s}{r^4}}.$$

(35)

Substitute (35) into Equation (33) to obtain

$$\begin{array}{cc}& \gamma R^{\prime }+\gamma w{R}^{″}-2(3R+207w{R}^{\prime }+6w^2{R}^{″}-160w^3{R}^{‴}-32w^4{R}^{‴})\hfill \\ & +3(R^2+20wR{R}^{\prime }-8w^{2}R{R}^{″}+56w^2{R}^{\prime 2}-32w^3{R}^{\prime }{R}^{″})=0.\hfill \end{array}$$

(36)

3. Power Series Solutions and Proof of Convergence

In this section, we examine a nonlinear ordinary differential equation (NLODE) that is known to be complex and cannot be solved using basic functions and integrals. However, employing the power series method [44] can effectively address such intricate ordinary differential problems. Here, we present the power series solution and verify the convergence of this solution.

We define that Equation (36) has a power series solution of the following form:

$$R\left(w\right)={\displaystyle \sum _{n=0}^{\infty }}{a}_n{w}^n,$$

(37)

so that

$$\begin{array}{cc}& R^{\prime }\left(w\right)={\displaystyle \sum _{n=0}^{\infty }}(n+1)a_{n+1}{w}^n,\hfill \\ & R^{″}\left(w\right)={\displaystyle \sum _{n=0}^{\infty }}(n+1)(n+2)a_{n+2}{w}^n,\hfill \\ & R^{‴}\left(w\right)={\displaystyle \sum _{n=0}^{\infty }}(n+1)(n+2)(n+3)a_{n+3}{w}^n,\hfill \\ & R^{‴}\left(w\right)={\displaystyle \sum _{n=0}^{\infty }}(n+1)(n+2)(n+3)(n+4)a_{n+4}{w}^n.\hfill \end{array}$$

(38)

Equation (36) can be converted into the following form:

$$\begin{array}{c}\gamma {\displaystyle \sum _{n=0}^{\infty }}(n+1)a_{n+1}{w}^n+\gamma {\displaystyle \sum _{n=0}^{\infty }}n(n+1)a_{n+1}{w}^n-2(3{\displaystyle \sum _{n=0}^{\infty }}{a}_n{w}^n+207{\displaystyle \sum _{n=0}^{\infty }}(n+1)a_{n+1}{w}^n\\ +6{\displaystyle \sum _{n=0}^{\infty }}(n-1)n{a}_n{w}^n-160{\displaystyle \sum _{n=0}^{\infty }}(n-2)(n-1)n{a}_n{w}^n-32{\displaystyle \sum _{n=0}^{\infty }}(n-3)(n-2)(n-1)n{a}_n{w}^n)\\ +3({\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{k=0}^n}{a}_k{a}_{n-k}{w}^n+20{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{k=0}^n}(n-k)a_k{a}_{n-k}{w}^n-8{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{k=0}^n}(n-k)(n-k-1)a_k{a}_{n-k}{w}^n\\ +56{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{k=0}^n}k(n-k)a_k{a}_{n-k}{w}^n-32{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{k=0}^n}k(n-k)(n-k-1)a_k{a}_{n-k}{w}^n)=0.\end{array}$$

(39)

Based on the current conditions for $n=0,$ we obtain

$$a_1={\displaystyle \frac{6a_0-3a_0^2}{\gamma -414}},$$

(40)

and for $n\ge 1$,

$$\begin{array}{cc}\hfill a_{n+1}& ={\displaystyle \frac{1}{\gamma {(n+1)}^2}}[2(3a_n+207n{a}_{n+1}+6n(n-1)a_n-160n(n-1)(n-2)a_n-32n(n-1)(n-2)(n-3)a_n)\hfill \\ \hfill & -3({\displaystyle \sum _{k=0}^n}{a}_k{a}_{n-k}+20{\displaystyle \sum _{k=0}^n}(n-k)a_k{a}_{n-k}-8{\displaystyle \sum _{k=0}^n}(n-k)(n-k+1)a_k{a}_{n-k}+56{\displaystyle \sum _{k=0}^n}k(n-k)a_k{a}_{n-k}\hfill \\ \hfill & -32{\displaystyle \sum _{k=0}^n}k(n-k)(n-k-1)a_k{a}_{n-k}\left)\right].\hfill \end{array}$$

(41)

The power series solution of Equation (4) is

$$\begin{array}{cc}\hfill R\left(w\right)& =a_0+a_{1}w+{\displaystyle \sum _{n=1}^{\infty }}{a}_{n+1}{w}^{n+1}\hfill \\ \hfill & =a_0+{\displaystyle \frac{6a_0-3a_0^2}{\gamma -414}}w+{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{1}{\gamma {(n+1)}^2}}\left[2\right(3a_n+207n{a}_{n+1}+6n(n-1)a_n-160n(n-1)(n-2)a_n\hfill \\ \hfill & -32n(n-1)(n-2)(n-3)a_n)-3({\displaystyle \sum _{k=0}^n}{a}_k{a}_{n-k}+20{\displaystyle \sum _{k=0}^n}(n-k)a_k{a}_{n-k}-8{\displaystyle \sum _{k=0}^n}(n-k)(n-k+1)a_k{a}_{n-k}\hfill \\ \hfill & +56{\displaystyle \sum _{k=0}^n}k(n-k)a_k{a}_{n-k}-32{\displaystyle \sum _{k=0}^n}k(n-k)(n-k-1)a_k{a}_{n-k}\left)\right]w^{n+1}.\hfill \end{array}$$

(42)

The particular solution of Equation (4) is

$$\begin{array}{cc}\hfill u(x,y,z,t)& ={\displaystyle \frac{R\left(w\right)}{r}}+{\displaystyle \frac{1-\alpha -\beta }{6}}r\hfill \\ \hfill & ={\displaystyle \frac{1}{x+y-t}}\{a_0+{\displaystyle \frac{6a_0-3a_0^2}{\gamma -414}}{\displaystyle \frac{z^2-\gamma t^2}{{\left(x+y-t\right)}^4}}+{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{1}{\gamma {\left(n+1\right)}^2}}\left[2a_n(3a_n+207n{a}_{n+1}+6n(n-1)a_n\right.\hfill \\ \hfill & -160n(n-1)(n-2)a_n-32n(n-1)(n-2)(n-3)a_n)-3({\displaystyle \sum _{k=0}^n}{a}_k{a}_{n-k}+20{\displaystyle \sum _{k=0}^n}(n-k)a_k{a}_{n-k}\hfill \\ \hfill & -8{\displaystyle \sum _{k=0}^n}(n-k)(n-k+1)a_k{a}_{n-k}+56{\displaystyle \sum _{k=0}^n}k(n-k)a_k{a}_{n-k}\hfill \\ \hfill & -32{\displaystyle \sum _{k=0}^n}k(n-k)(n-k-1)a_k{a}_{n-k}\left)\right]{({\displaystyle \frac{z^2-\gamma t^2}{{\left(x+y-t\right)}^4}})}^{n+1}\}+{\displaystyle \frac{(1-\alpha -\beta )(x+y-t)}{6}}.\hfill \end{array}$$

(43)

Now, we can prove the convergence of the solution; it can be seen from the above formula that

$$\left|a_{n+1}\right|\le M\left[3\left|a_{n+1}\right|+4\left|a_n\right|+5{\displaystyle \sum _{k=0}^n}\left|a_k\right|\left|a_{k+1}\right|\right],n=0,1,2,\cdots$$

(44)

with $M=max\left\{\left|{\displaystyle \frac{414}{\gamma }}\right|,1\right\}.$ Then, define a new power series $\psi =P\left(z\right)={\sum }_{n=0}^{\infty }{p}_n{z}_n,$ where $p_0=\left|\begin{array}{c}{a}_0\end{array}\right|,p_1=\left|\begin{array}{c}{a}_1\end{array}\right|,p_{n+4}=M\left[\begin{array}{c}3\left|\begin{array}{c}{a}_{n+1}\end{array}\right|+4\left|\begin{array}{c}{a}_n\end{array}\right|+5{\sum }_{k=0}^n\left|\begin{array}{c}{a}_k\end{array}\right|\left|\begin{array}{c}{a}_{k+1}\end{array}\right|\end{array}\right],n=0,1,2,\cdots .$

It is easy to see that $\left|\begin{array}{c}{a}_n\end{array}\right|\le p_n,n=0,1,2,\cdots$. In other words, the series constructed above is an optimal series of the power series (43). Next, we need to demonstrate that the power series has a positive convergence radius by examining the series operation.

$$\begin{array}{cc}\hfill P\left(z\right)& =p_0+p_{1}z+{\displaystyle \sum _{n=1}^{\infty }}{p}_{n+1}{z}^{n+1}\hfill \\ \hfill & =p_0+p_{1}z+M{\displaystyle \sum _{n=1}^{\infty }}(3\left|a_{n+1}\right|+4\left|a_n\right|+5{\displaystyle \sum _{k=0}^n}\left|a_k\right|\left|a_{n-k}\right|)z^{n+1}\hfill \\ \hfill & =p_0+p_{1}z+M(3{\displaystyle \sum _{n=1}^{\infty }}{p}_{n+1}{z}^{n+1}+4{\displaystyle \sum _{n=1}^{\infty }}{p}_n{z}^{n+1}+5{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \sum _{k=0}^n}{p}_k{p}_{n-k}{z}^{n+1})\hfill \\ \hfill & =p_0+p_{1}z+M(3(P\left(z\right)-p_0-p_{1}z)+4z(P\left(z\right)-p_0)+5z(P^2\left(z\right)-p_0^2)).\hfill \end{array}$$

(45)

Now, we construct an implicit function of z:

$$F(z,\psi )=\psi -p_0-p_{1}z-M(3(P\left(z\right)-p_0-p_{1}z)+4z(P\left(z\right)-p_0)+5z(P^2\left(z\right)-p_0^2)).$$

(46)

Obviously, $F(z,\psi )$ is analytic, and $F(0,p_0)=0,F_{\psi }^{\prime }(0,p_0)\ne 0.$ According to the existence theorem of implicit function, the series $\psi =P\left(z\right)$ is analytic in the field, so there is a positive convergence radius. Therefore, the power series solution is convergent.

4. Solutions Using $({\mathit{G}}^{\prime }/{\mathit{G}}^{\mathbf{2}})$-Expansion Method

In this section, we will illustrate the application of the $(G^{\prime }/G^2)$-expansion method [23] to find exact solitary wave solutions of the KPB-like equation. By utilizing the traveling wave transformation $\xi =x+y+z-ct$, Equation (4) is transformed into the following ODE in the variable $U=U\left(\xi \right)$:

$$U^{″″}+6U^{\prime }{U}^{″}+(\alpha +\beta -\gamma -2c+c^2)U^{″}=0.$$

(47)

By integrating the equation with respect to $\xi$ once and then choosing the constant of integration to be zero, we obtain the following ODE:

$$U^{‴}+3{\left(U^{\prime }\right)}^2+(\alpha +\beta -\gamma -2c+c^2)U^{\prime }=0,$$

(48)

for which the homogeneous balance principle is applied. The highest-order derivative $U^{‴}$ and the nonlinear term of the highest order ${\left(U^{\prime }\right)}^2$ are balanced via

$$\mathrm{deg}\left[U^{‴}\right]=N+3=\mathrm{deg}\left[{\left(U^{\prime }\right)}^2\right]=2(N+1),$$

(49)

which leads to $N=1.$ Hence, the form of exact solutions of the ODE using the $(G^{\prime }/G^2)$-expansion method can be expressed as

$$U\left(\xi \right)=a_0+a_1\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)+b_1{\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)}^{-1},$$

(50)

where the unknown constants $a_0,a_1,b_1$ may be zero, but both of them cannot be zero simultaneously, and $G=G\left(\xi \right)$ satisfies the following NLODE:

$${\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)}^{\prime }=\mu +\lambda {\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)}^2$$

(51)

in which $\mu \ne 1\mathrm{and}\lambda \ne 0$ are integers. Substituting (50) into (48) along with (51), we obtain

$$\begin{array}{cc}\hfill & (2a_1\lambda {\mu }^2-2b_1\mu {\lambda }^2)+8a_1{\lambda }^2\mu {\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)}^2+6a_1{\lambda }^3{\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)}^4-6b_1{\mu }^3{\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)}^{-4}\hfill \\ \hfill & -8b_1{\mu }^2\lambda {\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)}^{-2}+3a_1^2{\mu }^2+3a_1^2{\lambda }^2{\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)}^4+3b_1^2{\mu }^2{\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)}^{-4}+3b_1^2{\lambda }^2\hfill \\ \hfill & +6a_1^2\mu \lambda {\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)}^2-6a_1{b}_1{\mu }^2{\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)}^{-2}-12a_1{b}_1\lambda \mu -6a_1{b}_1{\lambda }^2{\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)}^2\hfill \\ \hfill & +6b_1^2\mu \lambda {\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)}^{-2}+(\alpha +\beta -\gamma -2c+c^2)(a_1\mu -b_1\lambda +a_1\lambda {\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)}^2-b_1\mu {\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)}^{-2})=0.\hfill \end{array}$$

(52)

Then, collecting all the coefficients with the same power of ${(G^{\prime }/G^2)}^i,i=0,\pm 1,\pm 2,\cdots$, and finally setting these resulting coefficients to be zero, we obtain the following system of algebraic equations in $a_0,a_1,b_1,\mu ,\lambda ,c,\alpha ,\beta ,\gamma .$

$$\begin{array}{cc}\hfill & {\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)}^{-4}:-6b_1{\mu }^3+3b_1^2{\mu }^2=0,\hfill \\ \hfill & {\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)}^{-2}:-8b_1{\mu }^2\lambda -6a_1{b}_1{\mu }^2+6b_1^2\lambda \mu +(\alpha +\beta -\gamma -2c+c^2)(-b_1\mu )=0,\hfill \\ \hfill & {\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)}^0:2a_1\lambda {\mu }^2-2b_1\mu {\lambda }^2+3a_1^2{\mu }^2+3b_1^2{\lambda }^2-12a_1{b}_1\lambda \mu +(\alpha +\beta -\gamma -2c+c^2)(a_1\mu -b_1\lambda )=0,\hfill \\ \hfill & {\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)}^2:8a_1{\lambda }^2\mu +6a_1^2\mu \lambda -6a_1{b}_1{\lambda }^2+a_1\lambda (\alpha +\beta -\gamma -2c+c^2)=0,\hfill \\ \hfill & {\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)}^4:6a_1{\lambda }^3+3a_1^2{\lambda }^2=0.\hfill \end{array}$$

(53)

By solving the algebraic system using Maple 2024, we obtain three cases:

Case1

$$\begin{array}{cc}\hfill & a_0=a_0,\hfill \\ \hfill & a_1=0,\hfill \\ \hfill & b_1=2\mu ,\hfill \\ \hfill & c=-1\pm \sqrt{-4\lambda \mu +\alpha +\beta -\gamma +1},\hfill \end{array}$$

(54)

where $a_0,\alpha ,\beta ,\gamma ,\lambda ,\mu$ are arbitrary constants.

Case2

$$\begin{array}{cc}\hfill & a_0=a_0,\hfill \\ \hfill & a_1=-2\lambda ,\hfill \\ \hfill & b_1=0,\hfill \\ \hfill & c=-1\pm \sqrt{-4\lambda \mu +\alpha +\beta -\gamma +1},\hfill \end{array}$$

(55)

where $a_0,\alpha ,\beta ,\gamma ,\lambda ,\mu$ are arbitrary constants.

Case3

$$\begin{array}{cc}\hfill & a_0=a_0,\hfill \\ \hfill & a_1=-2\lambda ,\hfill \\ \hfill & b_1=2\mu ,\hfill \\ \hfill & c=-1\pm \sqrt{-16\lambda \mu +\alpha +\beta -\gamma +1},\hfill \end{array}$$

(56)

where $a_0,\alpha ,\beta ,\gamma ,\lambda ,\mu$ are arbitrary constants.

With these parameters, we can write three results of solutions to Equation (4) as follows.

Result1 (from Case1, we have $\xi =x+y+z+(1\mp \sqrt{-4\lambda \mu +\alpha +\beta -\gamma +1})t$). When $\mu \lambda >0$, the trigonometric function solution corresponding to the parameter values in Case1 can be written as

$$u^{\left(1\right)}=a_0+2\sqrt{\lambda \mu }\left({\displaystyle \frac{Dcos\left(\sqrt{\mu \lambda }\xi \right)-Csin\left(\sqrt{\mu \lambda }\xi \right)}{Ccos\left(\sqrt{\mu \lambda }\xi \right)+Dsin\left(\sqrt{\mu \lambda }\xi \right)}}\right).$$

(57)

When $\mu \lambda <0$, the exponential function solution associated with the parameter values in Case1 can be expressed as

$$u^{\left(2\right)}=a_0+4\lambda \mu {\left(2\sqrt{\left|\mu \lambda \right|}-{\displaystyle \frac{4C\sqrt{\left|\mu \lambda \right|}{e}^{2\xi \sqrt{\left|\mu \lambda \right|}}}{C{e}^{2\xi \sqrt{\left|\mu \lambda \right|}}-D}}\right)}^{-1}.$$

(58)

When $\mu =0,\lambda \ne 0,$ the exact solution corresponding to the parameter values in Case1 is $u^{\left(3\right)}=a_0$, which is the constant solution.

Result2 (from Case2, we have $\xi =x+y+z+(1\mp \sqrt{-4\lambda \mu +\alpha +\beta -\gamma +1})t$). When $\mu \lambda >0$, the trigonometric function solution corresponding to the parameter values in Case2 can be written as

$$u^{\left(4\right)}=a_0-2\sqrt{\lambda \mu }\left({\displaystyle \frac{Ccos\left(\sqrt{\mu \lambda }\xi \right)+Dsin\left(\sqrt{\mu \lambda }\xi \right)}{Dcos\left(\sqrt{\mu \lambda }\xi \right)-Csin\left(\sqrt{\mu \lambda }\xi \right)}}\right).$$

(59)

When $\mu \lambda <0$, the exponential function solution associated with the parameter values in Case2 can be expressed as

$$u^{\left(5\right)}=a_0-2\sqrt{\left|\mu \lambda \right|}+{\displaystyle \frac{4C\sqrt{\left|\mu \lambda \right|}{e}^{2\xi \sqrt{\left|\mu \lambda \right|}}}{C{e}^{2\xi \sqrt{\left|\mu \lambda \right|}}-D}}.$$

(60)

When $\mu =0,\lambda \ne 0,$ the rational function solution corresponding to the parameter values in Case2 can be expressed as

$$u^{\left(6\right)}=a_0+{\displaystyle \frac{2C}{C\xi +D}}.$$

(61)

Result3 (from Case3, we have $\xi =x+y+z+(1\mp \sqrt{-16\lambda \mu +\alpha +\beta -\gamma +1})t$). When $\mu \lambda >0$, the trigonometric function solution corresponding to the parameter values in Case3 can be written as

$$u^{\left(7\right)}=a_0-2\sqrt{\lambda \mu }\left({\displaystyle \frac{Ccos\left(\sqrt{\lambda \mu }\xi \right)+Dsin\left(\sqrt{\lambda \mu }\xi \right)}{Dcos\left(\sqrt{\lambda \mu }\xi \right)-Csin\left(\sqrt{\lambda \mu }\xi \right)}}\right)+2\sqrt{\lambda \mu }\left({\displaystyle \frac{Dcos\left(\sqrt{\lambda \mu }\xi \right)-Csin\left(\sqrt{\lambda \mu }\xi \right)}{Ccos\left(\sqrt{\lambda \mu }\xi \right)+Dsin\left(\sqrt{\lambda \mu }\xi \right)}}\right).$$

(62)

When $\mu \lambda <0$, the exponential function solution associated with the parameter values in Case3 can be expressed as

$$u^{\left(8\right)}=a_0-2\sqrt{\left|\lambda \mu \right|}+{\displaystyle \frac{4C\sqrt{\left|\lambda \mu \right|}{e}^{2\xi \sqrt{\left|\lambda \mu \right|}}}{C{e}^{2\xi \sqrt{\left|\lambda \mu \right|}}-D}}+4\lambda \mu {\left(2\sqrt{\left|\lambda \mu \right|}-{\displaystyle \frac{4C\sqrt{\left|\lambda \mu \right|}{e}^{2\xi \sqrt{\left|\lambda \mu \right|}}}{C{e}^{2\xi \sqrt{\left|\lambda \mu \right|}}-D}}\right)}^{-1}.$$

(63)

When $\mu =0,\lambda \ne 0,$ the rational function solution corresponding to the parameter values in Case3 can be expressed as

$$u^{\left(9\right)}=a_0+{\displaystyle \frac{2C}{C\xi +D}}.$$

(64)

5. Bilinear Auto-Bäcklund Transformation and Multiple-Soliton Solutions

5.1. Auto-Bäcklund Transformation

Through the following transformation:

$$u(x,y,z,t)=2{(lnf(x,y,z,t))}_x,$$

(65)

the bilinear form of Equation (4) gives

$$(D_x^3{D}_y+D_x{D}_t+D_y{D}_t+D_t^2+\alpha D_x^2+\beta D_y^2-\gamma D_z^2)f·f=0.$$

(66)

To derive a bilinear auto-Bäcklund transformation of Equation (4), we consider the transformation between the solution and another solution of the same bilinear equation when given the following form:

$$\begin{array}{cc}& ((D_x^3{D}_y+D_x{D}_t+D_y{D}_t+D_t^2+\alpha D_x^2+\beta D_y^2-\gamma D_z^2)g·g)f^2-g^2\left(\right(D_x^3{D}_y\hfill \\ & +D_x{D}_t+D_y{D}_t+D_t^2+\alpha D_x^2+\beta D_y^2-\gamma D_z^2)f·f)=0,\hfill \end{array}$$

(67)

where g is a function of $x,y,z,\mathrm{and}t$. According to the exchange formulas for the Hirota bilinear operators,

$$\begin{array}{cc}& (D_x{D}_{t}g·g)f^2-g^2(D_x{D}_{t}f·f)=2D_x(D_{t}g·f)·fg,\hfill \\ & (D_x^{2}g·g)f^2-g^2(D_x^{2}f·f)=2D_x(D_{x}g·f)·fg,\hfill \\ & (D_x^3{D}_{y}g·g)f^2-g^2(D_x^3{D}_{y}f·f)=2D_y(D_x^{3}g·f)·fg+6D_x(D_x{D}_{y}g·f)·(D_{x}f·g).\hfill \end{array}$$

(68)

Equation (4) can be rewritten as

$$\begin{array}{cc}\hfill 0& =(D_x^3{D}_{y}g·g)f^2-g^2(D_x^3{D}_{y}f·f)+(D_x{D}_{t}g·g)f^2-g^2(D_x{D}_{t}f·f)\hfill \\ \hfill & +(D_y{D}_{t}g·g)f^2-g^2(D_y{D}_{t}f·f)+(D_t^{2}g·g)f^2-g^2(D_t^{2}f·f)\hfill \\ \hfill & +(\alpha D_x^{2}g·g)f^2-g^2(\alpha D_x^{2}f·f)+(\beta D_y^{2}g·g)f^2-g^2(\beta D_y^{2}f·f)\hfill \\ \hfill & -\gamma D_z^{2}g·g)f^2+g^2(\gamma D_z^{2}f·f)\hfill \\ \hfill & =2D_x((D_t+\alpha D_x)g·f)·gf+2D_y((D_x^3+\beta D_y)g·f)·gf+2D_z(-\gamma D_{z}g·f)·fg\hfill \\ \hfill & +6D_x(D_x{D}_{y}g·f)·(D_{x}f·g)+2D_t((D_t+D_y)g·f)·gf.\hfill \end{array}$$

(69)

Hence, we obtain a bilinear auto-Bäcklund transformation for Equation (4) as

$$\begin{array}{cc}\hfill & (D_t+\alpha D_x)g·f={\upsilon }_{1}gf,\hfill \\ \hfill & (D_x^3+\beta D_y)g·f={\upsilon }_{2}gf,\hfill \\ \hfill & D_x{D}_{y}g·f={\upsilon }_3{D}_{x}f·g,\hfill \\ \hfill & (D_t+D_y)g·f={\upsilon }_{4}gf,\hfill \\ \hfill & -\gamma D_{z}g·f={\upsilon }_{5}gf,\hfill \end{array}$$

(70)

where ${\upsilon }_1,{\upsilon }_2,{\upsilon }_3,{\upsilon }_4,\mathrm{and}{\upsilon }_5$ are all constants.

Choose $f=1$ as a solution of the bilinear form, corresponding to $u=2{(lnf)}_x=0,$ and substitute $f=1$ into the auto-Bäcklund transformation; we thus have

$$\begin{array}{cc}\hfill & g_t+\alpha g_x=0,\hfill \\ \hfill & g_{xxx}+\beta g_y=0,\hfill \\ \hfill & g_{xy}={\upsilon }_3{g}_x,\hfill \\ \hfill & g_t+g_y=0,\hfill \\ \hfill & -\gamma g_z=0.\hfill \end{array}$$

(71)

Set the exponential function of $g=1+\delta e^{{\rho }_{1}x+{\rho }_{2}y+{\rho }_{3}z+{\rho }_{4}t}$ as the bilinear form solution, where $\delta ,{\rho }_1,{\rho }_2,{\rho }_3,and{\rho }_4$ are all constants. Let ${\upsilon }_1={\upsilon }_2={\upsilon }_4={\upsilon }_5=0$ to obtain

$${\rho }_3=0,{\rho }_2={\upsilon }_3,\alpha =-{\displaystyle \frac{{\rho }_4}{{\rho }_1}},\beta =-{\displaystyle \frac{{\rho }_1^3}{{\upsilon }_3}}.$$

(72)

Therefore, we derive the corresponding exponential function solution for Equation (4) as

$$u=2{(lng)}_x={\displaystyle \frac{2{\rho }_1\delta e^{{\rho }_{1}x+{\nu }_{3}y+{\rho }_{4}t}}{1+\delta e^{{\rho }_{1}x+{\nu }_{3}y+{\rho }_{4}t}}}.$$

(73)

5.2. Multiple-Soliton Solutions

In this section, we seek to derive the dispersion relation for the new extended KPB-like equation by applying Hirota’s direct method [16] and considering the phase shift of soliton interactions to obtain multiple-soliton solutions. We start by substituting

$$u(x,y,z,t)=e^{{\theta }_i},{\theta }_i=k_{i}x+r_{i}y+s_{i}z-w_{i}t$$

(74)

into the linear terms of Equation (4), and this in turn gives the dispersion relations as

$$w_i={\displaystyle \frac{k_i+r_i\pm \sqrt{k_i^2+r_i^2+2k_i{r}_i-4k_i^3{r}_i-4\alpha k_i^2-4\beta r_i^2+4\gamma s_i^2}}{2}}$$

(75)

with $k_i^2+r_i^2+2k_i{r}_i-4k_i^3{r}_i-4\alpha k_i^2-4\beta r_i^2+4\gamma s_i^2\ge 0,i=1,2,\cdots ,N.$ Consequently, the phase variables are as follows:

$${\theta }_i=k_{i}x+r_{i}y+s_{i}z-{\displaystyle \frac{k_i+r_i\pm \sqrt{k_i^2+r_i^2+2k_i{r}_i-4k_i^3{r}_i-4\alpha k_i^2-4\beta r_i^2+4\gamma s_i^2}}{2}}t,i=1,2,\cdots ,N.$$

(76)

Next, we use the transformation

$$u(x,y,z,t)=2{(lnf(x,y,z,t))}_x$$

(77)

into Equation (4), where the auxiliary function $f(x,y,z,t)$ for the single-soliton solution is given as

$$f(x,y,z,t)=1+e^{{\theta }_1}=1+e^{k_{1}x+r_{1}y+s_{1}z-{\displaystyle \frac{k_1+r_1\pm \sqrt{k_1^2+r_1^2+2k_1{r}_1-4k_1^3{r}_1-4\alpha k_1^2-4\beta r_1^2+4\gamma s_1^2}}{2}}t}.$$

(78)

Using (78) into (77) leads to the single-soliton solution below:

$$u(x,y,z,t)={\displaystyle \frac{2k_1{e}^{k_{1}x+r_{1}y+s_{1}z-\frac{k_1+r_1\pm \sqrt{k_1^2+r_1^2+2k_1{r}_1-4k_1^3{r}_1-4\alpha k_1^2-4\beta r_1^2+4\gamma s_1^2}}{2}t}}{1+e^{k_{1}x+r_{1}y+s_{1}z-\frac{k_1+r_1\pm \sqrt{k_1^2+r_1^2+2k_1{r}_1-4k_1^3{r}_1-4\alpha k_1^2-4\beta r_1^2+4\gamma s_1^2}}{2}t}}}.$$

(79)

For the two-soliton solution, we use the auxiliary function as

$$f(x,y,z,t)=1+e^{{\theta }_1}+e^{{\theta }_2}+a_{12}{e}^{{\theta }_1+{\theta }_2},$$

(80)

where $a_{12}$ is the phase shift of the interaction of solitons. To determine the phase shift $a_{12}$, we substitute (80) into Equation (4): $a_{12}=-{\displaystyle \frac{P(p_1-p_2)}{P(p_1+p_2)}},$ and

$$\begin{array}{c}P(p_1-p_2)={(k_1-k_2)}^3(r_1-r_2)+(k_1-k_2)(-w_1+w_2)+(r_1-r_2)(-w_1+w_2)\\ +{(w_1-w_2)}^2+\alpha {(k_1-k_2)}^2+\beta {(r_1-r_2)}^2-\gamma {(s_1-s_2)}^2.\end{array}$$

(81)

$$\begin{array}{c}P(p_1+p_2)={(k_1+k_2)}^3(r_1+r_2)+(k_1+k_2)(-w_1-w_2)+(r_1+r_2)(-w_1-w_2)\\ +{(w_1+w_2)}^2+\alpha {(k_1+k_2)}^2+\beta {(r_1+r_2)}^2-\gamma {(s_1+s_2)}^2.\end{array}$$

(82)

Solving for the phase shift $a_{12}$ gives

$$a_{12}={\displaystyle \frac{3k_1{k}_2(k_1-k_2)(r_1-r_2)+k_1^3{r}_2+k_2^3{r}_1+2\alpha k_1{k}_2+2\beta r_1{r}_2-2\gamma s_1{s}_2+\frac{1}{2}{A}_1{A}_2-\frac{1}{2}(k_1+r_1)(k_2+r_2)}{3k_1{k}_2(k_1+k_2)(r_1+r_2)+k_1^3{r}_2+k_2^3{r}_1+2\alpha k_1{k}_2+2\beta r_1{r}_2-2\gamma s_1{s}_2+\frac{1}{2}{A}_1{A}_2-\frac{1}{2}(k_1+r_1)(k_2+r_2)}},$$

(83)

and this gives the general phase shift $a_{ij}$

$$a_{ij}={\displaystyle \frac{3k_i{k}_j(k_i-k_j)(r_i-r_j)+k_i^3{r}_j+k_j^3{r}_i+2\alpha k_i{k}_j+2\beta r_i{r}_j-2\gamma s_i{s}_j+\frac{1}{2}{A}_i{A}_j-\frac{1}{2}(k_i+r_i)(k_j+r_j)}{3k_i{k}_j(k_i+k_j)(r_i+r_j)+k_i^3{r}_j+k_j^3{r}_i+2\alpha k_i{k}_j+2\beta r_i{r}_j-2\gamma s_i{s}_j+\frac{1}{2}{A}_i{A}_j-\frac{1}{2}(k_i+r_i)(k_j+r_j)}},$$

(84)

where

$$\begin{array}{c}{A}_i=\sqrt{{\left(k_i+r_i\right)}^2-4(k_i^3{r}_i+\alpha k_i^2+\beta r_i^2-\gamma s_i^2)},\\ A_j=\sqrt{{(k_j+r_j)}^2-4(k_j^3{r}_j+\alpha k_j^2+\beta r_j^2-\gamma s_j^2)},\\ 1\le i<j\le N.\end{array}$$

(85)

The two-soliton solution is obtained by substituting (80), (83), and (85) into Equation (77).

For the three-soliton solution, we apply the auxiliary function $f(x,y,z,t)$ as

$$f(x,y,z,t)=1+e^{{\theta }_1}+e^{{\theta }_2}+e^{{\theta }_3}+a_{12}{e}^{{\theta }_1+{\theta }_2}+a_{13}{e}^{{\theta }_1+{\theta }_3}+a_{23}{e}^{{\theta }_2+{\theta }_3}+b_{123}{e}^{{\theta }_1+{\theta }_2+{\theta }_3},$$

(86)

where

$$b_{123}=a_{12}{a}_{13}{a}_{23}.$$

(87)

The three-soliton solution is obtained by substituting (86) into Equation (77). This also shows that N-soliton solutions can be obtained for finite N, where $N\ge 1.$

6. Display the Solutions Obtained Graphically

To illustrate the properties of the solution more effectively, we present relevant graphics with the help of Maple. Three-dimensional plots provide a spatial representation of the solution’s distribution, allowing for the identification of both local characteristics and global trends. This visualization reveals how the solution changes across the entire space. Additionally, 2D contour plots clearly depict the contours of the solution, facilitating an understanding of its gradient and local features. Two-dimensional density graphics further enhance this understanding by illustrating the density distribution of the data. This representation helps to uncover correlations between the two variables and identifies areas of concentration and sparsity within the data. Furthermore, the solution we obtained includes numerous arbitrary constants, which can describe the internal structure and various characteristics of the waves [45,46,47,48,49].

In the following figures, we present the 3D plots, 2D density plots, and 2D contour plots of some exact solutions, which are $u^{\left(7\right)},u^{\left(8\right)},\mathrm{and}{u}^{\left(9\right)}$, expressed in (62), (63), and (64), respectively. They use the appropriate sets of parameter values selected below to expand the description. Employing $a_0=0,C=D=1,$ and the parameter values $\lambda =0.2,\mu =0.8,\alpha =\beta =2,\gamma =1,$ and choosing $\xi =x+2.2t,\xi =x-0.2t,$ we obtain the plots of the selected exact traveling wave solution as shown in Figure 1, which represents the trigonometric function solution. Now, we customary $a_0=C=D=1,\lambda =-0.2, \mu =0.8,\alpha =\beta =2,\gamma =1,$ under the condition $\lambda \mu <0$, and then (63) converts into the exponential function solution. From the solutions of (63), the exponential function solution is obtained as shown in Figure 2. If we set $a_0=1,\lambda =0.5,\mu =0,\alpha =\beta =0.1,\gamma =1,C=2,D=1$ in (64), then the rational function solution is instigated in Figure 3.

After selecting the appropriate parameter conditions, the physical structures of the single-soliton, double-soliton, and three-soliton solutions are illustrated in Figure 4, Figure 5, and Figure 6, respectively. Each figure comprises three subgraphs: subgraph (a) displays the corresponding 3D plot, subgraph (b) presents the 2D density plot, and subgraph (c) presents the 2D contour plot. Figure 4 reveals that the single-soliton maintains a relatively smooth profile during transmission, with minimal changes in amplitude and waveform. It flows in a positive direction over time. In contrast, Figure 5 demonstrates that the double solitons gradually separate following their interaction, continuing to move in the positive direction and slowly forming distinct waveforms. Figure 6 indicates that the motion of the original soliton alters after interaction, highlighting the transformation and transmission of information and energy during this process. Collectively, these observations enhance our understanding of the characteristics of soliton solutions and the nonlinear wave behavior described by the KPB-like equation.

7. Conclusions

This paper presents an analysis of a newly extended (3+1)-dimensional KPB-like equation, focusing on the derivation of various exact solutions. It can contribute to a deeper understanding of the general laws and complex physical phenomena associated with this equation. Initially, the integrability of the equation is established through the Painlevé analysis method. Subsequently, the vector field and Lie symmetry group are derived using the Lie symmetry method, leading to the reduction of the equation to ODEs. A power series solution is obtained, demonstrating favorable convergence properties. By employing the $(G^{\prime }/G^2)$-expansion method alongside the bilinear method, we derive trigonometric, exponential, rational function solutions, and multiple-soliton solutions. The exact solutions obtained can describe the propagation characteristics of light, with their variables representing the distribution of light intensity, phase, and other properties in space and time. Furthermore, the generated solutions facilitate the study of wave interactions across various physical structures and high-dimensional systems. To provide a more intuitive understanding of the partial solutions, we present corresponding 3D and 2D visualizations that illustrate the inherent characteristics of these solutions. In the future, in order to better understand the KPB-like equation, we can study other properties of the equation, and use the new methods to obtain different types of exact solutions.