Analytic Investigation of a Generalized Variable-Coefficient KdV Equation with External-Force Term

Abstract

This paper investigates integrable properties of a generalized variable-coefficient Korteweg–de Vries (gvcKdV) equation incorporating dissipation, inhomogeneous media, and an external-force term. Based on Painlevé analysis, sufficient and necessary conditions for the equation’s Painlevé integrability are obtained. Under specific integrability conditions, the Lax pair for this equation is successfully constructed using the extended Ablowitz–Kaup–Newell–Segur system (AKNS system). Furthermore, the Riccati-type Bäcklund transformation (R-BT), Wahlquist–Estabrook-type Bäcklund transformation (WE-BT), and the nonlinear superposition formula are derived. In utilizing these transformations and the formula, explicit one-soliton-like and two-soliton-like solutions are constructed from a seed solution. Moreover, the infinite conservation laws of the equation are systematically derived. Finally, the influence of variable coefficients and the external-force term on the propagation characteristics of a solitory wave is discussed, and soliton interaction is illustrated graphically.

1. Introduction

In research on solitary wave theory, the study of nonlinear evolution equations (NLEEs) with complete integrability has been a key focus in describing a wide variety of physical phenomena in the fields of hydro-dynamics, ocean dynamics, and plasma physics [1,2,3,4,5]. However, classical NLEEs with constant coefficients may exhibit limitations in specific problems. When dealing with inhomogeneous media or nonuniform boundaries, NLEEs with variable coefficients can more effectively capture the complex nonlinear phenomena involved [6,7]. Variable-coefficient KdV (vcKdV) equations and their extended forms are highly valued for modeling wave phenomena in specific physical settings, such as those involving changes in depth, width, or external fields, and their crucial role has been demonstrated previously across diverse applications including plasma physics, signal processing, and chemical transport [4,8,9,10,11].

This paper focuses on investigating the following generalized variable-coefficient KdV (gvcKdV) equation, which includes dispersion, perturbation, and an external-force term:

$$u_t+\left[{\alpha }_0\left(t\right)+{\alpha }_1\left(t\right)x\right]u_x+{\alpha }_2\left(t\right)u{u}_x+{\alpha }_3\left(t\right)u_{xxx}+{\alpha }_4\left(t\right)u=h\left(t\right),$$

(1)

where $u(x,t)$ represents the wave amplitude, a function of the spatial variable x and the temporal variable t. Coefficients ${\alpha }_i\left(t\right)(i=0,1,\cdots ,4)$[12] and $h\left(t\right)$ are real analytic functions dependent on time t, which, respectively, correspond to perturbation, the nonuniformity of media, the effects of the nonlinearity, third-order dispersion, dissipation, and external force. The external-force term $h\left(t\right)$ can effectively simulate background forcing and external driving in physical systems, such as fluid-filled elastic tubes and internal solitary waves [7,13,14].

Equation (1) and its special cases can be employed to describe problems arising in diverse physical contexts, such as the propagation of weakly nonlinear waves in shallow water tunnels of variable depth [15], the evolution of internal gravity waves in lakes with varying cross-sections [16], and ion-acoustic solitary waves in plasma physics. The variable coefficients, dissipation, perturbation, and external-force term within the equation enable a more refined characterization of the complex influences arising from temporal and spatial variations, as well as external driving effects, in real physical systems

In various types of plasmas and fluids, several special cases of Equation (1) have been obtained:

• When ${\alpha }_0\left(t\right)=4d\left(t\right),{\alpha }_1\left(t\right)=-b\left(t\right),{\alpha }_2\left(t\right)=6a\left(t\right),{\alpha }_3\left(t\right)=a\left(t\right),{\alpha }_4\left(t\right)=-2b\left(t\right)$, and $h\left(t\right)=0$, Equation (1) becomes

  $$u_t+6a\left(t\right)u{u}_x+a\left(t\right)u_{xxx}-b\left(t\right)(2u+x{u}_x)+4d\left(t\right)u_x=0,$$

  (2)

  which describes radially ingoing acoustic waves in cylindrical plasma, interfacial waves in two-layer liquids, or Alfvén waves in non-interacting plasma. Its Lax pair, infinite conservation laws, and various exact solutions were obtained in [17].
• When ${\alpha }_1\left(t\right)=0,{\alpha }_2\left(t\right)\ne 0,$ and ${\alpha }_3\left(t\right)\ne 0$, Equation (1) is simplified to

  $$u_t+{\alpha }_0\left(t\right)u_x+{\alpha }_2\left(t\right)u{u}_x+{\alpha }_3\left(t\right)u_{xxx}+{\alpha }_4\left(t\right)u=h\left(t\right).$$

  (3)
  This equation describes phenomena such as ISWs in the ocean, pressure pulses in fluid-filled tubes of special value in arterial dynamics, trapped quasi-one-dimensional Bose–Einstein condensates, and ion-acoustic solitary waves in plasmas. Its Lax pair, auto-Bäcklund transformation, nonlinear superposition formula, and an infinite number of conservation laws have been investigated in [18].
• In considering an artery as a prestressed, tapered, thin-walled, long, and circularly conical elastic tube, and blood as an incompressible inviscid fluid, the propagation of weakly nonlinear waves in such a fluid-filled elastic tube follows the following equation [19]:

  $$U_{\tau }+{\mu }_{1}U{U}_{\xi }+{\mu }_2{U}_{\xi \xi \xi }+A\tau {\mu }_3{U}_{\xi }=0,$$

  (4)

  where coefficients ${\mu }_1,{\mu }_2,$ and ${\mu }_3$ depend on the material of the tube and the initial deformation. When $u=U, t=\tau , x=\xi ,{\alpha }_0\left(t\right)=A\tau {\mu }_3, {\alpha }_2\left(t\right)={\mu }_1, {\alpha }_3\left(t\right)={\mu }_2,$ and ${\alpha }_1\left(t\right)={\alpha }_4\left(t\right)=h\left(t\right)=0$, Equation (4) is a special case of Equation (1).

In this paper, Section 2 will extend the Painlevé test to Equation (1) in order to derive constraints on variable coefficients under the Painlevé property and establish necessary and sufficient conditions for Painlevé integrability. Section 3 will construct a Lax pair with the AKNS system, along with Riccati-type (R-BT) and Wahlquist–Estabrook-type (WE-BT) Bäcklund transformations under specific Painlevé conditions. Furthermore, one-soliton-like and two-soliton-like solutions will be investigated with R-BT and the nonlinear superposition formula based on WE-BT. Infinite conservation laws for Equation (1) will be obtained in Section 4. Section 5 will provide a substantial analysis of the propagation of one-soliton-like and two-soliton-like solutions for Equation (1) and discussions on the effects of the variable coefficients and the external-force term on given solutions. Finally, Section 6 will conclude the paper.

2. Painlevé Integrability

A partial differential equation is said to possess the Painlevé property if its solutions are single-valued in relation to the singular manifolds of noncharacteristic flows. The Painlevé test is a useful method for investigating the integrability of a given nonlinear equation [20]. It is conjectured that equations possessing Painlevé integrability exhibit many important properties, such as being solvable using the inverse scattering method and admitting Lax pairs, Bäcklund transformations, N-soliton solutions, and infinite conservation laws [21]. Building upon established Painlevé analysis frameworks in [22,23,24,25], this paper extends these techniques to the gvcKdV equation with an external-force term.

We assume that the solution $u(x,t)$ of Equation (1) can be expressed in terms of a generalized Laurent series

$$u(x,t)={\varphi }^{-p}(x,t)\sum _{j=0}^{\infty }{u}_j(x,t){\varphi }^j(x,t),$$

(5)

and adopt Kruskal’s simplified ansatz [20]

$$\varphi (x,t)=x+\psi \left(t\right),$$

(6)

where $u_j(x,t)$ (for $j=0,1,2,\cdots$ with $u_0\ne 0$) and $\psi \left(t\right)$ are analytic functions, $\varphi (x,t)=0$ defines the noncharacteristic movable singularity manifold, and p is a positive integer. We assume ${\alpha }_1\left(t\right)\ne 0$. Equation (1) is then rewritten as

$$u_t+[{\alpha }_0\left(t\right)+{\alpha }_1\left(t\right)\varphi (x,t)-\psi \left(t\right){\alpha }_1\left(t\right)]u_x+{\alpha }_2\left(t\right)u{u}_x+{\alpha }_3\left(t\right)u_{xxx}+{\alpha }_4\left(t\right)u=h\left(t\right).$$

(7)

By leading-order analysis, we immediately obtain

$$p=2,u_0(x,t)=-{\displaystyle \frac{12{\alpha }_3\left(t\right)}{{\alpha }_2\left(t\right)}}{\varphi }_x^2.$$

(8)

Substituting Laurent series (5) into Equation (7) and setting the coefficients of all powers of $\varphi$ to be zero, specifically for ${\varphi }^{j-5}$, we obtain via symbolic computation that

$$\begin{array}{cc}\hfill {\varphi }^{j-5}(j\ne 5):& u_{j-3,t}+(j-4)u_{j-2}{\varphi }_t+\left({\alpha }_0-\psi {\alpha }_1\right)u_{j-3,x}+\left({\alpha }_0-\psi {\alpha }_1\right)(j-4)u_{j-2}{\varphi }_x\hfill \\ \hfill & +{\alpha }_1{u}_{j-4,x}+{\alpha }_1(j-5)u_{j-3}{\varphi }_x+{\alpha }_2\sum _{k=0}^j\left[u_{k-1,x}+(k-2)u_k{\varphi }_x\right]u_{j-k}\hfill \\ \hfill & +{\alpha }_3(j-2)(j-3)(j-4)u_j{\varphi }_x^3+3{\alpha }_3(j-3)(j-4)u_{j-1,x}{\varphi }_x^2\hfill \\ \hfill & +3{\alpha }_3(j-3)(j-4)u_{j-1}{\varphi }_x{\varphi }_{xx}+{\alpha }_3(j-4)u_{j-2}{\varphi }_{xxx}+{\alpha }_3{u}_{j-3,xxx}\hfill \\ \hfill & +3{\alpha }_3(j-4)u_{j-2,xx}{\varphi }_x+3{\alpha }_3(j-4)u_{j-2,x}{\varphi }_{xx}+{\alpha }_4{u}_{j-3}=0,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\varphi }^{j-5}(j=5):& u_{j-3,t}+(j-4)u_{j-2}{\varphi }_t+\left({\alpha }_0-\psi {\alpha }_1\right)u_{j-3,x}+\left({\alpha }_0-\psi {\alpha }_1\right)(j-4)u_{j-2}{\varphi }_x\hfill \\ \hfill & +{\alpha }_1{u}_{j-4,x}+{\alpha }_1(j-5)u_{j-3}{\varphi }_x+{\alpha }_2\sum _{k=0}^j\left[u_{k-1,x}+(k-2)u_k{\varphi }_x\right]u_{j-k}\hfill \\ \hfill & +{\alpha }_3(j-2)(j-3)(j-4)u_j{\varphi }_x^3+3{\alpha }_3(j-3)(j-4)u_{j-1,x}{\varphi }_x^2\hfill \\ \hfill & +3{\alpha }_3(j-3)(j-4)u_{j-1}{\varphi }_x{\varphi }_{xx}+{\alpha }_3(j-4)u_{j-2}{\varphi }_{xxx}+{\alpha }_3{u}_{j-3,xxx}\hfill \\ \hfill & +3{\alpha }_3(j-4)u_{j-2,xx}{\varphi }_x+3{\alpha }_3(j-4)u_{j-2,x}{\varphi }_{xx}+{\alpha }_4{u}_{j-3}-h=0.\hfill \end{array}$$

(10)

For $j=0$, the expression for $u_0$ ensures that the coefficient of ${\varphi }^{-5}$ vanishes. For $j>0$, the coefficient of $u_j$ in the resulting equation is

$$-2{\alpha }_2\left(t\right)u_0{\varphi }_x+{\alpha }_2\left(t\right)(j-2)u_0{\varphi }_x+{\alpha }_3\left(t\right)(j-2)(j-3)(j-4){\varphi }_x^3.$$

(11)

In substituting the expression for $u_0$, this coefficient becomes

$${\alpha }_3\left(t\right)(j+1)(j-4)(j-6){\varphi }_x^3.$$

(12)

Since ${\alpha }_3\left(t\right)\ne 0$, we obtain the recursion relation

$$(j+1)(j-4)(j-6)u_j=F_j,$$

(13)

where $F_j$ represents the terms involving $u_k$ with $k<j$.

The resonances occur at $j=-1,4,6$. The resonance at $j=-1$ corresponds to the arbitrariness of the singularity manifold $\varphi (x,t)$. We now check the compatibility conditions at the resonances $j=4$ and $j=6$.

With the Kruskal ansatz $\varphi (x,t)=x+\psi \left(t\right)$, the calculations for the first few $u_j$ yield

$$\begin{array}{cc}\hfill & j=1:u_1={\displaystyle \frac{12{\alpha }_3\left(t\right)}{{\alpha }_2\left(t\right)}}{\varphi }_{xx}=0(\mathrm{condition}\mathrm{satisfied}),\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & j=2:u_2=-{\displaystyle \frac{1}{{\alpha }_2\left(t\right)}}({\psi }^{\prime }\left(t\right)+{\alpha }_0\left(t\right)-\psi \left(t\right){\alpha }_1\left(t\right))\Rightarrow u_{2,x}=0,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & j=3:u_3=-{\displaystyle \frac{1}{{\alpha }_2\left(t\right)}}\left({\displaystyle \frac{{{\alpha }_2}^{\prime }\left(t\right)}{{\alpha }_2\left(t\right)}}-{\displaystyle \frac{{{\alpha }_3}^{\prime }\left(t\right)}{{\alpha }_3\left(t\right)}}+2{\alpha }_1\left(t\right)-{\alpha }_4\left(t\right)\right)\Rightarrow u_{3,x}=0,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & j=4:\mathrm{Compatibility}\mathrm{condition}\mathrm{satisfied}\mathrm{identically},u_4\mathrm{is}\mathrm{arbitrary},\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & j=5:u_5={\displaystyle \frac{1}{6{\alpha }_3\left(t\right)}}(u_{2,t}+{\alpha }_4\left(t\right)u_2-h\left(t\right))\Rightarrow u_{5,x}=-u_{4,xx},\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & j=6:u_{3,t}+({\alpha }_1\left(t\right)+{\alpha }_4\left(t\right))u_3+{\alpha }_2\left(t\right)u_3^2=0.\hfill \end{array}$$

(19)

Since the resonances are at $j=-1,4,6$, for Equation (1) to possess the Painlevé property, the compatibility conditions at $j=4$ and $j=6$ must hold. The conditions derived at $j=3$ and $j=6$ must be simultaneously satisfied.

Substituting the expression for $u_3\left(t\right)$ from (16) into condition (19) yields the following constraint on the coefficients:

$$\begin{array}{cc}\hfill & {\alpha }_3\left(t\right){\alpha }_2\left(t\right){\alpha }_2^{″}\left(t\right)-{\alpha }_2{\left(t\right)}^2{\alpha }_3^{″}\left(t\right)+2{\alpha }_3\left(t\right){\alpha }_2{\left(t\right)}^2{\alpha }_1^{\prime }\left(t\right)+4{\alpha }_3\left(t\right){\alpha }_4\left(t\right){\alpha }_2\left(t\right){\alpha }_2^{\prime }\left(t\right)\hfill \\ \hfill & -5{\alpha }_1\left(t\right){\alpha }_3\left(t\right){\alpha }_2\left(t\right){\alpha }_2^{\prime }\left(t\right)+3{\alpha }_2\left(t\right){\alpha }_2^{\prime }\left(t\right){\alpha }_3^{\prime }\left(t\right)-3{\alpha }_3\left(t\right){\alpha }_2^{\prime }{\left(t\right)}^2+3{\alpha }_1\left(t\right){\alpha }_2{\left(t\right)}^2{\alpha }_3^{\prime }\left(t\right)\hfill \\ \hfill & -3{\alpha }_4\left(t\right){\alpha }_2{\left(t\right)}^2{\alpha }_3^{\prime }\left(t\right)-{\alpha }_3\left(t\right){\alpha }_2{\left(t\right)}^2{\alpha }_4^{\prime }\left(t\right)-2{\alpha }_3\left(t\right){\alpha }_4{\left(t\right)}^2{\alpha }_2{\left(t\right)}^2\hfill \\ \hfill & -2{\alpha }_1{\left(t\right)}^2{\alpha }_3\left(t\right){\alpha }_2{\left(t\right)}^2+5{\alpha }_1\left(t\right){\alpha }_3\left(t\right){\alpha }_4\left(t\right){\alpha }_2{\left(t\right)}^2=0.\hfill \end{array}$$

(20)

It can be observed that condition (20) for Equation (1) to possess the Painlevé property is independent of the external forcing term $h\left(t\right)$, which is consistent with the results obtained for the unforced equation [12]. Solving this constraint equation leads to several sets of sufficient conditions for Painlevé integrability. Three sets for the case $h\left(t\right)=0$ are as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}{\alpha }_3\left(t\right)=\hfill & k{\alpha }_2\left(t\right)+c_1{\alpha }_2\left(t\right)\times \left(\int {\alpha }_2\left(t\right){\mathrm{e}}^{-3\int {\alpha }_1\left(t\right)\mathrm{d}t}\mathrm{d}t\right),\hfill \\ {\alpha }_4\left(t\right)=\hfill & 2{\alpha }_1\left(t\right),\hfill \end{array}\right.\end{array}\end{array}$$

(21)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}{\alpha }_3\left(t\right)=\hfill & k{\alpha }_2\left(t\right),\hfill \\ {\alpha }_4\left(t\right)=\hfill & 2{\alpha }_1\left(t\right)\mathrm{or}\hfill \\ {\alpha }_4\left(t\right)=\hfill & {\displaystyle \frac{{\alpha }_2\left(t\right){\mathrm{e}}^{-\int 3{\alpha }_1\left(t\right)\mathrm{d}t}}{c_2+2\int \left({\alpha }_2\left(t\right){\mathrm{e}}^{-\int 3{\alpha }_1\left(t\right)\mathrm{d}t}\right)\mathrm{d}t}}+2{\alpha }_1\left(t\right),\hfill \end{array}\right.\end{array}\end{array}$$

(22)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}{\alpha }_3\left(t\right)=\hfill & c_3{\alpha }_1\left(t\right){\mathrm{e}}^{\int {\alpha }_1\left(t\right)\mathrm{d}t},\hfill \\ {\alpha }_4\left(t\right)=\hfill & {\displaystyle \frac{{\alpha }_2^{\prime }\left(t\right)}{{\alpha }_2\left(t\right)}}-{\displaystyle \frac{{\alpha }_1^{\prime }\left(t\right)}{{\alpha }_1\left(t\right)}},\hfill \end{array}\right.\end{array}\end{array}$$

(23)

where $k,c_1,c_2,$ and $c_3$ are arbitrary constants.

We now derive a necessary and sufficient condition for Equation (1) to be Painlevé integrable, following a similar approach as in [26]. This requires satisfying both conditions (16) and (19). We analyze condition (19) by considering cases for $u_3\left(t\right)$.

Case 1: $u_3\left(t\right)=0$. In this case, condition (19) is trivially satisfied. Condition (16) then becomes an equation for ${\alpha }_3\left(t\right)$:

$$\begin{array}{cc}\hfill u_3\left(t\right)=0& \iff {\displaystyle \frac{{{\alpha }_2}^{\prime }\left(t\right)}{{\alpha }_2\left(t\right)}}-{\displaystyle \frac{{{\alpha }_3}^{\prime }\left(t\right)}{{\alpha }_3\left(t\right)}}+2{\alpha }_1\left(t\right)-{\alpha }_4\left(t\right)=0,\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & \iff {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}ln\left|{\displaystyle \frac{{\alpha }_3\left(t\right)}{{\alpha }_2\left(t\right)}}\right|=2{\alpha }_1\left(t\right)-{\alpha }_4\left(t\right),\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & \iff {\alpha }_3\left(t\right)={\alpha }_2\left(t\right){\mathrm{e}}^{\int (2{\alpha }_1\left(t\right)-{\alpha }_4\left(t\right))\mathrm{d}t}.\hfill \end{array}$$

(26)

Case 2: $u_3\left(t\right)\ne 0$. In this case, Equation (19) is a Bernoulli equation for $u_3\left(t\right)$, whose solution is

$$u_3\left(t\right)={\displaystyle \frac{{\mathrm{e}}^{-\int ({\alpha }_1\left(t\right)+{\alpha }_4\left(t\right))\mathrm{d}t}}{c+\int {\alpha }_2\left(t\right){\mathrm{e}}^{-\int ({\alpha }_1\left(t\right)+{\alpha }_4\left(t\right))\mathrm{d}t}\mathrm{d}t}},$$

(27)

where c is an arbitrary constant. Substituting this non-zero $u_3\left(t\right)$ back into expression (16), we obtain ${\alpha }_3\left(t\right)$:

$${\alpha }_3\left(t\right)={\alpha }_2\left(t\right){\mathrm{e}}^{\int (2{\alpha }_1\left(t\right)-{\alpha }_4\left(t\right)+{\alpha }_2\left(t\right)u_3\left(t\right))\mathrm{d}t}.$$

(28)

In combining both cases, the necessary and sufficient condition for Equation (1) to pass the Painlevé test requires coefficients ${\alpha }_i\left(t\right)$ to satisfy the condition:

$${\alpha }_3\left(t\right)={\alpha }_2\left(t\right){\mathrm{e}}^{\int 2{\alpha }_1\left(t\right)-{\alpha }_4\left(t\right)\mathrm{d}t}(a+b{\mathrm{e}}^{\int ({\displaystyle \frac{{\alpha }_2\left(t\right){\mathrm{e}}^{-\int ({\alpha }_1\left(t\right)+{\alpha }_4\left(t\right))\mathrm{d}t}}{c+\int {\alpha }_2\left(t\right){\mathrm{e}}^{-\int ({\alpha }_1\left(t\right)+{\alpha }_4\left(t\right))\mathrm{d}t}\mathrm{d}t}})\mathrm{d}t}),$$

(29)

where $ab=0$ and $a^2+b^2\ne 0$.

Remark 1.

Condition (29) encompasses the three sets of sufficient conditions (21)–(23) as special cases under specific choices of constants and coefficient relationships.

3. AKNS System and Soliton-like Solutions

3.1. Lax Pair

The Lax pair may assure the complete integrability of NLEEs [18,27]. To construct the Lax pair for Equation (1), we introduce functions $E(x,t)$ and $F(x,t)$ within the AKNS system. The Lax pair for Equation (1) can be expressed as follows [12]:

$$\begin{array}{cc}\hfill {\mathrm{\Phi }}_x=& U\mathrm{\Phi }=\left(\begin{array}{cc}\lambda & u+W\left(t\right)+E(x,t)\\ F(x,t)& -\lambda \end{array}\right)\mathrm{\Phi },\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill {\mathrm{\Phi }}_t=& V\mathrm{\Phi }=\left(\begin{array}{cc}A(x,t,\lambda )& B(x,t,\lambda )\\ C(x,t,\lambda )& -A(x,t,\lambda )\end{array}\right)\mathrm{\Phi },\hfill \end{array}$$

(31)

where function $\mathrm{\Phi }={({\varphi }_1,{\varphi }_2)}^T$, superscript T denotes the transpose of the matrix, U and V are two $2\times 2$ null-trace matrices, eigenvalue $\lambda$ is a parameter independent of x and t, and $A(x,t,\lambda )$, $B(x,t,\lambda )$, and $C(x,t,\lambda )$ are functions of x and t.

$A(x,t,\lambda )$, $B(x,t,\lambda )$, and $C(x,t,\lambda )$ are expanded with respect to $\lambda$ as follows:

$$\left\{\begin{array}{c}A(x,t,\lambda )=a_0(x,t)+a_1(x,t)\lambda +a_2(x,t){\lambda }^2+a_3(x,t){\lambda }^3,\hfill \\ B(x,t,\lambda )=b_0(x,t)+b_1(x,t)\lambda +b_2(x,t){\lambda }^2,\hfill \\ C(x,t,\lambda )=c_0(x,t)+c_1(x,t)\lambda +c_2(x,t){\lambda }^2,\hfill \end{array}\right.$$

(32)

By substituting them into the compatibility condition $U_t-V_x+[U,V]=0$, equating the coefficients of powers of $\lambda$, and utilizing the third set of Painlevé integrability conditions (23), we obtain the following via symbolic computation:

$$\begin{array}{cc}\hfill q\left(t\right)=& {\alpha }_4\left(t\right)-{\alpha }_1\left(t\right),p\left(t\right)={\displaystyle \frac{{\alpha }_0\left(t\right){\alpha }_1\left(t\right)}{{\alpha }_2\left(t\right)}},\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill M\left(t\right)=& {\displaystyle \frac{{\alpha }_1\left(t\right)}{{\alpha }_2\left(t\right)}},E(x,t)=M\left(t\right)x+Q\left(t\right),\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill Q\left(t\right)=& {\mathrm{e}}^{-\int q\left(t\right)\mathrm{d}t}\left(c_4-\int p\left(t\right){\mathrm{e}}^{\int q\left(t\right)\mathrm{d}t}\mathrm{d}t\right),\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill W\left(t\right)=& {\mathrm{e}}^{-\int q\left(t\right)\mathrm{d}t}\left(c_5-\int h\left(t\right){\mathrm{e}}^{\int \left(q\right(t\left)\right)\mathrm{d}t}\mathrm{d}t\right),\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill F(x,t)=& F\left(t\right)=-{\displaystyle \frac{{\alpha }_2\left(t\right)}{6{\alpha }_3\left(t\right)}},\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill A(x,t,\lambda )=& -4{\alpha }_3\left(t\right){\lambda }^3\hfill \\ \hfill & +\left[-{\alpha }_0\left(t\right)+{\displaystyle \frac{2}{3}}(Q\left(t\right)+W\left(t\right)){\alpha }_2\left(t\right)-{\displaystyle \frac{1}{3}}\left(u(x,t)+{\displaystyle \frac{x{\alpha }_1\left(t\right)}{{\alpha }_2\left(t\right)}}\right){\alpha }_2\left(t\right)\right]\lambda \hfill \\ \hfill & +\left[{\displaystyle \frac{{\alpha }_1\left(t\right)}{3}}-{\displaystyle \frac{{\alpha }_4\left(t\right)}{2}}-{\displaystyle \frac{1}{6}}{\alpha }_2\left(t\right)u_x\right],\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill & \hfill \\ \hfill B(x,t,\lambda )=& -4{\alpha }_3\left(t\right)\left(Q\left(t\right)+u(x,t)+W\left(t\right)+{\displaystyle \frac{x{\alpha }_1\left(t\right)}{{\alpha }_2\left(t\right)}}\right){\lambda }^2\hfill \\ \hfill & +\left(-2{\alpha }_3\left(t\right)\left({\displaystyle \frac{{\alpha }_1\left(t\right)}{{\alpha }_2\left(t\right)}}+u_x(x,t)\right)\right)\lambda -{\alpha }_3\left(t\right)u_{xx}\hfill \\ \hfill & +{\displaystyle \frac{1}{3}}\left(Q\left(t\right)+u(x,t)+W\left(t\right)+{\displaystyle \frac{x{\alpha }_1\left(t\right)}{{\alpha }_2\left(t\right)}}\right)\hfill \\ \hfill & \times \left(-{\alpha }_0\left(t\right)-{\displaystyle \frac{1}{3}}\left(u(x,t)+{\displaystyle \frac{x{\alpha }_1\left(t\right)}{{\alpha }_2\left(t\right)}}\right){\alpha }_2\left(t\right)+{\displaystyle \frac{2}{3}}(W\left(t\right)+Q\left(t\right)){\alpha }_2\left(t\right)\right),\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill C(x,t,\lambda )=& {\displaystyle \frac{2{\alpha }_2\left(t\right)}{3}}{\lambda }^2\hfill \\ \hfill & -{\displaystyle \frac{{\alpha }_2\left(t\right)}{6{\alpha }_3\left(t\right)}}\left(-{\alpha }_0\left(t\right)+{\displaystyle \frac{2}{3}}(W+Q){\alpha }_2\left(t\right)-{\displaystyle \frac{1}{3}}\left(u(x,t)+{\displaystyle \frac{x{\alpha }_1\left(t\right)}{{\alpha }_2\left(t\right)}}\right){\alpha }_2\left(t\right)\right).\hfill \end{array}$$

(40)

Remark 2.

Equations (30) and (31) with $c_4=h\left(t\right)=0$ are the Lax pair for Equation (1) with $h\left(t\right)=0$, which agrees with the corresponding results in [12].

3.2. Riccati Form and Auto-Bäcklund Transformation

The auto-Bäcklund transformation is a powerful tool for constructing solutions of NLEEs. Starting from a seed solution of an NLEE, one can construct other solutions, which often provides a transformation between the $(N-1)$-soliton solution and the N-soliton solution. To construct the auto-Bäcklund transformation for Equation (1) in the Riccati form, we introduce the function $\mathrm{\Gamma }(x,t)={\displaystyle \frac{{\varphi }_1}{{\varphi }_2}}$ [28]. Lax pair (30) and (31) can then be reduced to the following equivalent $\mathrm{\Gamma }$-Riccati system:

$$\begin{array}{cc}\hfill {\mathrm{\Gamma }}_x& =u+W\left(t\right)+M\left(t\right)x+Q\left(t\right)+2\lambda \mathrm{\Gamma }-F(x,t){\mathrm{\Gamma }}^2,\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill {\mathrm{\Gamma }}_t& =B(x,t,\lambda )+2A(x,t,\lambda )\mathrm{\Gamma }-C(x,t,\lambda ){\mathrm{\Gamma }}^2.\hfill \end{array}$$

(42)

The Riccati-type auto-Bäcklund transformation (R-BT) for Equation (1) is defined as follows:

$${\mathrm{\Gamma }}^{\prime }={\mathrm{\Gamma }}^{\prime }[\lambda ,X(x,t),\mathrm{\Gamma }],u^{\prime }=u+U[\lambda ,X(x,t),\mathrm{\Gamma }],$$

(43)

where ${\mathrm{\Gamma }}^{\prime }$ and $\mathrm{\Gamma }$ are two different solutions of $\mathrm{\Gamma }$-Riccati systems (41) and (42), $u^{\prime }$ and u are different solutions of Equation (1), and $X(x,t)$ is to be determined. With symbolic computation, substituting (43) into the spatial part (41) gives rise to the transformation relating $u^{\prime }$ and u via $\mathrm{\Gamma }$:

$$\begin{array}{cc}\hfill {\mathrm{\Gamma }}^{\prime }& =-{\displaystyle \frac{2\lambda }{F(x,t)}}-\mathrm{\Gamma },\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill u^{\prime }& =u-2{\mathrm{\Gamma }}_x.\hfill \end{array}$$

(45)

Equations (44) and (45) constitute the $\mathrm{\Gamma }$-R-BT for Equation (1), which also serves as a transformation between the $(N-1)$-soliton solution and the N-soliton solution.

Let $u=-{\omega }_x$ and $u^{\prime }=-{\omega }_x^{\prime }$, where ${\omega }^{\prime }(x,t)$ and $\omega (x,t)$ are potential functions. Solving Equation (45) for $\mathrm{\Gamma }$ yields

$$\mathrm{\Gamma }=\int {\displaystyle \frac{u-u^{\prime }}{2}}\mathrm{d}x+\mu \left(t\right)={\displaystyle \frac{1}{2}}\left({\omega }^{\prime }(x,t)-\omega (x,t)\right)+\mu \left(t\right),$$

(46)

where $\mu \left(t\right)$ is an arbitrary integration function of t. Substituting Equation (46) into $\mathrm{\Gamma }$-Riccati systems (41) and (42), we can obtain the Wahlquist–Estabrook-type auto-Bäcklund transformation (WE-BT) for Equation (1) as follows:

$$\begin{array}{cc}\hfill {\omega }_x^{\prime }-{\omega }_x& =2\left[-{\omega }_x+W\left(t\right)+E(x,t)+2\lambda \mathrm{\Gamma }-F(x,t){\mathrm{\Gamma }}^2\right],\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill {\omega }_t^{\prime }-{\omega }_t& =2\left[B(x,t,\lambda )+2A(x,t,\lambda )\mathrm{\Gamma }-C(x,t,\lambda ){\mathrm{\Gamma }}^2\right]-2{\mu }^{\prime }\left(t\right).\hfill \end{array}$$

(48)

Substituting $u=-{\omega }_x$ into Equation (1) yields the following [29]:

$$\begin{array}{c}\hfill {\omega }_t=-{\alpha }_0\left(t\right){\omega }_x-{\alpha }_1\left(t\right)x{\omega }_x+{\displaystyle \frac{{\alpha }_2\left(t\right)}{2}}{\omega }_x^2-{\alpha }_3\left(t\right){\omega }_{xxx}-({\alpha }_4\left(t\right)-{\alpha }_1\left(t\right))\omega -xh\left(t\right).\end{array}$$

(49)

From (48) and (49), we can derive expressions for ${\omega }_t^{\prime }+{\omega }_t$ and ${\omega }_t^{\prime }-{\omega }_t$:

$$\begin{array}{cc}\hfill {\omega }_t^{\prime }+{\omega }_t=& ({\omega }_t^{\prime }-{\omega }_t)+2{\omega }_t\hfill \\ \hfill =& 4{\lambda }^2{\alpha }_3\left(t\right)\left({\omega }_x-{\omega }_x^{\prime }\right)+4{\alpha }_3\left(t\right)\lambda \left(-{\displaystyle \frac{{\alpha }_1\left(t\right)}{{\alpha }_2\left(t\right)}}+{\omega }_{xx}\right)\hfill \\ \hfill & -{\alpha }_4\left(t\right)(\omega +{\omega }^{\prime })+{\displaystyle \frac{2}{3}}{\alpha }_1\left(t\right)(2\omega +{\omega }^{\prime })-{\alpha }_0\left(t\right)\left({\omega }_x+{\omega }_x^{\prime }\right)\hfill \\ \hfill & +{\displaystyle \frac{2}{3}}(2{\alpha }_1\left(t\right)-3{\alpha }_4\left(t\right))\mu \left(t\right)-2{\mu }^{\prime }\left(t\right)-{\displaystyle \frac{1}{3}}x{\alpha }_1\left(t\right)\left(5{\omega }_x+{\omega }_x^{\prime }\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{3}}{\alpha }_2\left(t\right){\omega }_x\left(2{\omega }_x+{\omega }_x^{\prime }\right)+{\displaystyle \frac{1}{3}}(-\omega +{\omega }^{\prime }+2\mu \left(t\right)){\alpha }_2\left(t\right){\omega }_{xx}\hfill \\ \hfill & -{\displaystyle \frac{2}{3}}(Q\left(t\right){\alpha }_2\left(t\right)+W\left(t\right))\left({\omega }_x-{\omega }_x^{\prime }\right)-2xh\left(t\right),\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill {\omega }_t^{\prime }-{\omega }_t=& 4{\lambda }^2{\alpha }_3\left(t\right)\left({\omega }_x-{\omega }_x^{\prime }\right)+4{\alpha }_3\left(t\right)\lambda \left(-{\displaystyle \frac{{\alpha }_1\left(t\right)}{{\alpha }_2\left(t\right)}}+{\omega }_{xx}\right)\hfill \\ \hfill & -{\alpha }_4\left(t\right)({\omega }^{\prime }-\omega )+{\displaystyle \frac{2}{3}}{\alpha }_1\left(t\right)({\omega }^{\prime }-\omega )-{\alpha }_0\left(t\right)\left({\omega }_x^{\prime }-{\omega }_x\right)\hfill \\ \hfill & +{\displaystyle \frac{2}{3}}(2{\alpha }_1\left(t\right)-3{\alpha }_4\left(t\right))\mu \left(t\right)-2{\mu }^{\prime }\left(t\right)-{\displaystyle \frac{1}{3}}x{\alpha }_1\left(t\right)\left({\omega }_x^{\prime }-{\omega }_x\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{3}}{\alpha }_2\left(t\right){\omega }_x\left({\omega }_x^{\prime }-{\omega }_x\right)+{\displaystyle \frac{2}{3}}W\left(t\right)({\omega }_x^{\prime }-{\omega }_x)+{\displaystyle \frac{1}{3}}({\omega }^{\prime }-\omega +2\mu \left(t\right)){\alpha }_2\left(t\right){\omega }_{xx}\hfill \\ \hfill & +{\displaystyle \frac{2}{3}}Q\left(t\right){\alpha }_2\left(t\right)\left({\omega }_x^{\prime }-{\omega }_x\right)+2{\alpha }_3\left(t\right){\omega }_{xxx},\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill {\omega }_x^{\prime }+{\omega }_x=& ({\omega }_x^{\prime }-{\omega }_x)+2{\omega }_x=2\left(W\left(t\right)+E(x,t)+2\lambda \mathrm{\Gamma }-F(x,t){\mathrm{\Gamma }}^2\right).\hfill \end{array}$$

(52)

In particular, in choosing $\mu \left(t\right)={\displaystyle \frac{\lambda }{F\left(t\right)}}$, Equations (50)–(52) can be simplified to

$$\begin{array}{cc}\hfill {\omega }_x^{\prime }+{\omega }_x=& 2W\left(t\right)+2M\left(t\right)x+2Q\left(t\right)+{\displaystyle \frac{2{\lambda }^2}{F\left(t\right)}}-F\left(t\right){\displaystyle \frac{{({\omega }^{\prime }-\omega )}^2}{2}},\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill {\omega }_t^{\prime }+{\omega }_t=& \left(4{\alpha }_3\left(t\right){\lambda }^2-{\displaystyle \frac{2}{3}}{\alpha }_2\left(t\right)Q\left(t\right)-{\displaystyle \frac{2}{3}}{\alpha }_2\left(t\right)W\left(t\right)\right)\left({\omega }_x-{\omega }_x^{\prime }\right)\hfill \\ \hfill & +{\displaystyle \frac{2}{3}}(2\omega +{\omega }^{\prime }){\alpha }_1\left(t\right)-{\alpha }_4\left(t\right)(\omega +{\omega }^{\prime })-{\alpha }_0\left(t\right)\left({\omega }_x+{\omega }_x^{\prime }\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{3}}x{\alpha }_1\left(t\right)\left(5{\omega }_x+{\omega }_x^{\prime }\right)+{\displaystyle \frac{1}{3}}{\alpha }_2\left(t\right){\omega }_x\left(2{\omega }_x+{\omega }_x^{\prime }\right)+{\displaystyle \frac{1}{3}}{\alpha }_2\left(t\right)({\omega }^{\prime }-\omega ){\omega }_{xx}-2xh\left(t\right),\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill {\omega }_t^{\prime }-{\omega }_t=& 4{\lambda }^2{\alpha }_3\left(t\right)\left({\omega }_x-{\omega }_x^{\prime }\right)-{\alpha }_4\left(t\right)({\omega }^{\prime }-\omega )\hfill \\ \hfill & +{\displaystyle \frac{2}{3}}{\alpha }_1\left(t\right)({\omega }^{\prime }-\omega )-{\alpha }_0\left(t\right)\left({\omega }_x^{\prime }-{\omega }_x\right)-{\displaystyle \frac{1}{3}}x{\alpha }_1\left(t\right)\left({\omega }_x^{\prime }-{\omega }_x\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{3}}{\alpha }_2\left(t\right){\omega }_x\left({\omega }_x^{\prime }-{\omega }_x\right)+{\displaystyle \frac{2}{3}}W\left(t\right)({\omega }_x^{\prime }-{\omega }_x)+{\displaystyle \frac{1}{3}}{\alpha }_2\left(t\right)({\omega }^{\prime }-\omega ){\omega }_{xx}\hfill \\ \hfill & +{\displaystyle \frac{2}{3}}Q\left(t\right){\alpha }_2\left(t\right)\left({\omega }_x^{\prime }-{\omega }_x\right)+2{\alpha }_3\left(t\right){\omega }_{xxx}.\hfill \end{array}$$

(55)

3.3. Nonlinear Superposition Formula

The nonlinear superposition formula of an NLEE allows one to obtain new, more complex solutions from known solutions. Via the WE-BT given by (53) and (54), the nonlinear superposition formula for Equation (1) can be derived [30].

Let ${\omega }_1$ and ${\omega }_2$ be two new solutions obtained from a seed solution ${\omega }_0$ corresponding to parameters ${\lambda }_1$ and ${\lambda }_2$ in Bäcklund transformation (53), respectively. Let ${\omega }_3$ be the solution obtained from ${\omega }_1$ with parameter ${\lambda }_2$, and ${\omega }_4$ be the solution obtained from ${\omega }_2$ with parameter ${\lambda }_1$. Then, we have

$$\begin{array}{cc}\hfill {\omega }_{1x}+{\omega }_{0x}=& 2W\left(t\right)+2M\left(t\right)x+2Q\left(t\right)+{\displaystyle \frac{2{\lambda }_1^2}{F}}-F{\displaystyle \frac{{({\omega }_1-{\omega }_0)}^2}{2}},\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill {\omega }_{2x}+{\omega }_{0x}=& 2W\left(t\right)+2M\left(t\right)x+2Q\left(t\right)+{\displaystyle \frac{2{\lambda }_2^2}{F}}-F{\displaystyle \frac{{({\omega }_2-{\omega }_0)}^2}{2}},\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill {\omega }_{3x}+{\omega }_{1x}=& 2W\left(t\right)+2M\left(t\right)x+2Q\left(t\right)+{\displaystyle \frac{2{\lambda }_2^2}{F}}-F{\displaystyle \frac{{({\omega }_3-{\omega }_1)}^2}{2}},\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill {\omega }_{4x}+{\omega }_{2x}=& 2W\left(t\right)+2M\left(t\right)x+2Q\left(t\right)+{\displaystyle \frac{2{\lambda }_1^2}{F}}-F{\displaystyle \frac{{({\omega }_4-{\omega }_2)}^2}{2}}.\hfill \end{array}$$

(59)

According to the theorem of permutability, we have ${\omega }_3={\omega }_4$. Subtracting Equation (56) from (58), and subtracting Equation (57) from (59), we obtain the following two expressions, which must be equal:

$$\begin{array}{cc}\hfill {\omega }_{3x}-{\omega }_{0x}=& {\displaystyle \frac{2}{F}}({\lambda }_2^2-{\lambda }_1^2)-{\displaystyle \frac{F}{2}}\left[{({\omega }_3-{\omega }_1)}^2-{({\omega }_1-{\omega }_0)}^2\right],\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill {\omega }_{4x}-{\omega }_{0x}=& {\displaystyle \frac{2}{F}}({\lambda }_1^2-{\lambda }_2^2)-{\displaystyle \frac{F}{2}}\left[{({\omega }_4-{\omega }_2)}^2-{({\omega }_2-{\omega }_0)}^2\right].\hfill \end{array}$$

(61)

Setting ${\omega }_3={\omega }_4$ and substracting Equation (60) from Equation (61) yield the nonlinear superposition formula for Equation (1):

$${\omega }_3={\omega }_0+{\displaystyle \frac{4({\lambda }_2^2-{\lambda }_1^2)({\omega }_{1x}-{\omega }_{2x})}{F^2{({\omega }_1-{\omega }_2)}^2}}.$$

(62)

3.4. Soliton-like Solutions

Choosing $u=-W\left(t\right)-M\left(t\right)x-Q\left(t\right)$ as a seed solution, from the spatial part of $\mathrm{\Gamma }$-Riccati system (41), one can obtain soliton-like solutions for Equation (1). Let $P(x,t)={\omega }^{\prime }(x,t)-\omega (x,t)$. By selecting different choices for the integration function, we can obtain the following two distinct forms for $P(x,t)$:

$$\begin{array}{c}\hfill P_{11}(x,t)={\displaystyle \frac{2\lambda }{F\left(t\right)}}·{\displaystyle \frac{e^{2(\lambda x+r(t\left)\right)}-1}{e^{2(\lambda x+r(t\left)\right)}+1}}={\displaystyle \frac{2\lambda }{F\left(t\right)}}tanh(\lambda x+r\left(t\right)),\end{array}$$

(63)

$$\begin{array}{c}\hfill P_{12}(x,t)={\displaystyle \frac{2\lambda }{F\left(t\right)}}·{\displaystyle \frac{e^{2(\lambda x+r(t\left)\right)}+1}{e^{2(\lambda x+r(t\left)\right)}-1}}={\displaystyle \frac{2\lambda }{F\left(t\right)}}coth(\lambda x+r\left(t\right)).\end{array}$$

(64)

Substituting (63) and (64) into temporal WE-BT Equation (54) (or (55)) allows us to solve for the phase function $r\left(t\right)$. This process yields two types of one-soliton-like solutions for Equation (1):

$$\begin{array}{cc}\hfill u_{11}(x,t)=& -M\left(t\right)x-Q\left(t\right)-W\left(t\right)-{\displaystyle \frac{2{\lambda }^2}{F\left(t\right)}}{sech}^2(\lambda x+r\left(t\right)),\hfill \end{array}$$

(65)

$$\begin{array}{cc}\hfill u_{12}(x,t)=& -M\left(t\right)x-Q\left(t\right)-W\left(t\right)+{\displaystyle \frac{2{\lambda }^2}{F\left(t\right)}}{csch}^2(\lambda x+r\left(t\right)),\hfill \end{array}$$

(66)

where $r\left(t\right)$ is given by

$$r\left(t\right)=\int \left[-\lambda {\alpha }_0\left(t\right)-4c_3{e}^{\int {\alpha }_1\left(t\right)\mathrm{d}t}{\lambda }^3{\alpha }_1\left(t\right)+\lambda (Q\left(t\right)+W\left(t\right)){\alpha }_2\left(t\right)\right]\mathrm{d}t+c_6,$$

(67)

where $c_6$ is an arbitrary constant.

Taking $u_{11}=-{\omega }_{1x}$ and $u_{12}=-{\omega }_{2x}$, and substituting them into nonlinear superposition Formula (62), we can obtain the two-soliton-like solution for the equation:

$$u_3(x,t)=-{\omega }_{3x}=-W\left(t\right)-M\left(t\right)x-Q\left(t\right)+{\displaystyle \frac{2({\lambda }_1^2-{\lambda }_2^2)\left({\lambda }_1^2{sech}^2{\xi }_1+{\lambda }_2^2{csch}^2{\xi }_2\right)}{F\left(t\right){\left({\lambda }_{1}tanh{\xi }_1-{\lambda }_{2}coth{\xi }_2\right)}^2}},$$

(68)

where ${\xi }_1={\lambda }_{1}x+r_1\left(t\right)$ and ${\xi }_2={\lambda }_{2}x+r_2\left(t\right)$, with $r_1\left(t\right)$ and $r_2\left(t\right)$ given by (67) using ${\lambda }_1$ and ${\lambda }_2$, respectively.

If the above procedure continues, in principle, the N-soliton-like ($N\ge 3$) solutions for Equation (1) can also be constructed using the nonlinear superposition formula.

4. Infinite Conservation Laws

In soliton studies, for isospectral Lax integrable nonlinear evolution equations, conservation laws can be constructed directly from their Lax pairs and the Riccati equations associated with the corresponding spectral problems [31,32]. For nonisospectral NLEEs, infinite conservation laws can also be derived by reconsidering the nonisospectral problems as new isospectral ones. In this section, we will utilize the previously presented auto-Bäcklund transformation, specifically expressions (50) and (52), to determine an infinite number of conservation laws for Equation (1).

To obtain the infinite conservation laws for Equation (1), one can generally proceed as follows [33]: Firstly, by introducing the variable $G=F\left(t\right)({\omega }^{\prime }-\omega )$ into (50) and (52), we obtain

$$G_x-2F(u+W\left(t\right)+M\left(t\right)x+Q)-2{\lambda }^2+{\displaystyle \frac{1}{2}}{G}^2=0,$$

(69)

$$G_t+{\left\{-G\left[-{\alpha }_{2}u-x{\alpha }_1+{\displaystyle \frac{2}{3}}{\alpha }_2\left(u+W\left(t\right)+M\left(t\right)x+Q\right)\right]+{\alpha }_{0}G+4{\lambda }^2{\alpha }_{3}G+2{\alpha }_{3}F{u}_x\right\}}_x=0.$$

(70)

If we denote $\mathrm{\Phi }=-{\alpha }_{2}u-x{\alpha }_1+{\displaystyle \frac{2}{3}}{\alpha }_2\left(u+W\left(t\right)+M\left(t\right)x+Q\right)$, the above equation becomes

$$G_t+{\left(-G\mathrm{\Phi }+4{\lambda }^2{\alpha }_3\left(t\right)G+2{\alpha }_3\left(t\right)F\left(t\right)u_x+{\alpha }_0\left(t\right)G\right)}_x=0.$$

(71)

Let $T=G$ and $X=-G\mathrm{\Phi }+4{\lambda }^2{\alpha }_3\left(t\right)G+2{\alpha }_3\left(t\right)F\left(t\right)u_x+{\alpha }_0\left(t\right)G$. Equation (71) gives a conservation law for the Equation (1) in the form $T_t+X_x=0$.

To recursively derive an infinite number of conservation laws, we expand G in the following form:

$$G=2\lambda +\sum _{m=1}^{\infty }{g}_m(x,t){\lambda }^{-m},$$

(72)

where $g_0=0$. The $g_m$ coefficients are polynomials in u, with partial derivatives with respect to x, and possibly x itself. Substituting this series into Equation (69) and equating the coefficients of ${\lambda }^{-m}$ to zero yields the recursion formula

$$g_m=-{\displaystyle \frac{1}{2}}{g}_{m-1,x}-{\displaystyle \frac{1}{4}}\sum _{j=1}^{m-1}{g}_j{g}_{m-1-j}m\ge 2,$$

(73)

where the first few terms can be listed as

$$\begin{array}{cc}\hfill g_1(x,t)& =F\left(t\right)\left(u+W\left(t\right)+M\left(t\right)x+Q\left(t\right)\right),\hfill \\ \hfill g_2(x,t)& =-{\displaystyle \frac{1}{2}}F\left(t\right)\left(u_x+M\left(t\right)\right),\hfill \\ \hfill g_3(x,t)& ={\displaystyle \frac{1}{4}}F\left(t\right)u_{xx}-{\displaystyle \frac{1}{4}}F{\left(t\right)}^2{\left(u+W\left(t\right)+M\left(t\right)x+Q\left(t\right)\right)}^2,\hfill \\ \hfill g_4(x,t)& =-{\displaystyle \frac{1}{8}}F\left(t\right)u_{xxx}+{\displaystyle \frac{1}{2}}F{\left(t\right)}^2\left(u+W\left(t\right)+M\left(t\right)x+Q\left(t\right)\right)\left(u_x+M\left(t\right)\right).\hfill \end{array}$$

(74)

With symbolic computation, substituting series (72) into Equation (71) leads to an infinite number of conservation laws for Equation (1), as follows,

$${\left(g_m\right)}_t+{\left(-g_m\mathrm{\Phi }+4{\alpha }_3\left(t\right)g_{m+2}+{\alpha }_0\left(t\right)g_m\right)}_x=0,m=1,2,3,\cdots .$$

(75)

For example, the first three conservation laws for Equation (1) can be explicitly expressed as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill T_1& =g_1=F\left(t\right)\left(u+W\left(t\right)+M\left(t\right)x+Q\left(t\right)\right),\hfill \\ \hfill X_1& =-g_1\mathrm{\Phi }+{\alpha }_0\left(t\right)g_1+4{\alpha }_3\left(t\right)g_3,\hfill \end{array}\end{array}$$

(76)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill T_2& =g_2=-{\displaystyle \frac{1}{2}}F\left(t\right)\left(u_x+M\left(t\right)\right),\hfill \\ \hfill X_2& =-g_2\mathrm{\Phi }+4{\alpha }_3\left(t\right)g_4+{\alpha }_0\left(t\right)g_2,\hfill \end{array}\end{array}$$

(77)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill T_3& =g_3={\displaystyle \frac{1}{4}}F\left(t\right)u_{xx}-{\displaystyle \frac{1}{4}}F{\left(t\right)}^2{\left(u+W\left(t\right)+M\left(t\right)x+Q\left(t\right)\right)}^2,\hfill \\ \hfill X_3& =-g_3\mathrm{\Phi }+4{\alpha }_3\left(t\right)g_5+{\alpha }_0\left(t\right)g_3.\hfill \end{array}\end{array}$$

(78)

In principle, we can exhibit an infinite number of conservation laws with symbolic computation. It should be noted that typically only the odd powers of $\lambda$ provide meaningful independent conservation laws, while the conservation laws derived from even powers are related to the derivatives of those from odd powers or are trivial.

5. Discussion

This section discusses the propagation and interaction characteristics of the one-soliton-like and two-soliton-like solutions for Equation (1). Under constraint condition (23), Equation (1) primarily depends on the three time-dependent coefficient functions ${\alpha }_0\left(t\right),{\alpha }_1\left(t\right),$ and ${\alpha }_2\left(t\right)$ and the external-force term $h\left(t\right)$, which are crucial for the equation to exhibit favorable properties. To understand how these variable coefficients and $h\left(t\right)$ influence the propagation of solitary waves within nonuniform backgrounds, we present a qualitative analysis and graphical illustrations of the obtained solutions.

For the one-soliton-like solution (65), the corresponding amplitude $A_w$, velocity $V_w$ and background of the solitary wave are given by

$$\begin{array}{cc}\hfill A_w=& {\displaystyle \frac{2{\lambda }^2}{\left|F\right(t\left)\right|}}=12{\lambda }^2\left|{\displaystyle \frac{c_3{\alpha }_1\left(t\right){\mathrm{e}}^{\int {\alpha }_1\left(t\right)\mathrm{d}t}}{{\alpha }_2\left(t\right)}}\right|,\hfill \end{array}$$

(79)

$$\begin{array}{cc}\hfill V_w=& {\alpha }_0\left(t\right)+4c_3{e}^{\int {\alpha }_1\left(t\right)\mathrm{d}t}{\lambda }^2{\alpha }_1\left(t\right)-(Q\left(t\right)+W\left(t\right)){\alpha }_2\left(t\right),\hfill \end{array}$$

(80)

$$\begin{array}{cc}\hfill B_w=& -M\left(t\right)x-Q\left(t\right)-W\left(t\right).\hfill \end{array}$$

(81)

It is observed that amplitude $A_w$ is determined by parameter $c_3,\lambda$, and coefficient functions ${\alpha }_1\left(t\right)$ and ${\alpha }_2\left(t\right)$. Velocity $V_w$ jointly depends on the parameters, the three coefficient functions ${\alpha }_0\left(t\right)$, ${\alpha }_1\left(t\right)$, and ${\alpha }_2\left(t\right)$, and the external-force term $h\left(t\right)$. Both the external-force term and ${\alpha }_0\left(t\right)$ affect the solitonic velocity and background significantly.

By selecting different parameter values, we illustrate the behavior of the one-soliton-like solution (65) and the two-soliton-like solution (68) graphically. In all subfigures, the color intensity represents the magnitude of the solution $u(x,t)$, where warmer colors (e.g., yellow/red) indicate higher values and cooler colors (e.g., white/blue) indicate lower values within the specified PlotRange of u indicated on the vertical axis of each subplot. From Figure 1 and

Table 1. Effects of the coefficients and the forcing term.

	Change in the Perturbed Coefficient	Change in the Inhomogeneity Coefficient	Change in the Nonlinearity Coefficient	Change in the Forcing Term
Solitonic amplitude	Not affected	Affected	Affected	Not affected
Solitonic velocity	Affected	Affected	Affected	Affected
Solitonic background	Affected	Affected	Affected	Affected

, one can observe the propagation of the one-soliton-like solution and the changes in its amplitude, velocity, and background resulting from variations in ${\alpha }_0\left(t\right),{\alpha }_1\left(t\right),$ and forcing term $h\left(t\right)$. Figure 2 displays the interaction of the two-soliton-like solution, i.e., a bright–dark hybrid soliton-like wave propagating in non-constant backgrounds. In Figure 2a, the faster wave (larger ${\lambda }_i$) overtakes the slower one (smaller ${\lambda }_i$), resulting in an elastic collision; in (b–f), the change in the perturbed coefficient and change in the external-force term have similar effects on velocity and background when $c_4=c_5=0$ via a comparison with Figure 1b–f. As evident from (79) and (80), the integral term ${\mathrm{e}}^{\int {\alpha }_1\left(t\right)\mathrm{d}t}$ significantly impacts both the amplitude and velocity. This time dependence of the wave parameters, driven by the variable coefficients, and the background term are the reasons these are termed “soliton-like” solutions [34,35,36].

6. Conclusions

In this study, we systematically investigated a type of generalized variable-coefficient Korteweg–de Vries (gvcKdV) equation that incorporates dissipation, external forcing, and a spatially dependent coefficient affecting the convective term. This equation is relevant for describing physical phenomena such as shallow water waves with varying depth and ion-acoustic waves in nonuniform plasmas. We applied Painlevé analysis and gave the conditions for the equation’s Painlevé integrability, including both sufficient and necessary criteria. Under specific integrability conditions, the Lax pair was successfully constructed based on the extended AKNS system. Consequently, the WE-BT and the associated nonlinear superposition formula were derived. In utilizing these transformations and the formula, explicit one-soliton-like and two-soliton-like solutions were constructed from a chosen seed solution. Furthermore, based on the Riccati equation associated with the Bäcklund transformation, an infinite sequence of conservation laws for the equation was systematically derived. Finally, concise graphs were obtained to analyze the propagation characteristics of one-soliton-like and two-soliton-like solutions for Equation (1), while the critical role of the time-dependent coefficients and the external-force term in soliton behavior and collisions were discussed.