Research paper
Breather, lump and N-soliton wave solutions of the (2+1)-dimensional coupled nonlinear partial differential equation with variable coefficients

https://doi.org/10.1016/j.cnsns.2021.106098Get rights and content

Highlights

  • In this paper, we mainly investigate a (2+1)-dimensional coupled nonlinear partial differential equation with variable coefficients in an inhomogeneous medium.

  • Based on the Hirota bilinear form and symbolic computation, the breather wave solutions and lump solutions are constructed by using the extended homoclinic breather technique and the generalized positive quadratic function method.

  • In Section 5, we add the N-soliton solution of the studied equation, and analyze the dynamic propagation behavior of one-soliton solution and two-soliton solution of the equation.

  • When the coefficients of the equation are different, the corresponding improved results are obtained for some special equations.

Abstract

In this paper, we mainly investigate a (2+1)-dimensional coupled nonlinear partial differential equation with variable coefficients in an inhomogeneous medium. Based on the Hirota bilinear form and symbolic computation, the breather wave solutions and lump solutions are constructed by using the extended homoclinic breather technique and the generalized positive quadratic function method. Also, Hirota bilinear method is applied to considered equation for finding N-soliton wave solutions. When the coefficients of the equation are different, the corresponding improved results are obtained for some special equations. Furthermore, by plotting the images of different types of solutions, their dynamic behaviors are analyzed.

Introduction

Nonlinear partial differential equations (NLPDEs) have been widely used in solving some complex problems in a variety of sciences and engineering, which can describe many nonlinear phenomena in plasmas physics, optical communication, quantum mechanics and other fields [1], [2], [3], [4], [5], [6]. There exist many significant methods to find the analytical solutions of NLPDEs, including inverse scattering [7], Bäcklund transformation [8], Painlevé analysis [9], Darboux transformation [10], Hirota bilinear method [11], and so on. These methods are used to investigate integrability and different analytical solutions for NLPDEs, such as soliton solutions, multi-wave solutions, lump solutions, breather solutions and rogue wave solutions have been obtained [12], [13], [14], [15], [16], [17].

A (2+1)-dimensional coupled nonlinear partial differential equation has been studied in Ref. [18] uxt+32uxuxx+14uxxxx+ρ1wx+ρ2uxy+ρ3uxx+ρ44(uxxxy+3uxuxy+3uxxuy)+ρ54(wxxy+3uxyw+3uywx)+ρ6(3wwx+12wxyy)=0,wx=uyy,where ρi,i=1,2,,6 are arbitrary constants. Bilinear forms, two solitary waves and lump waves of Eq. have been derived via the truncated Painlevé analysis, Hirota method and positive quadratic function approach. By combining an additional exponential function with a positive quadratic function, interaction solution between a lump and a single-soliton solution has been constructed [18].

On the other hand, when considering the inhomogeneities of media and nonuniformities of boundaries, the NLPDEs with variable coefficients can describe more realistic problems concerned with time varying environment, inhomogeneous medium, etc. To our knowledge, many different kinds of NLPDEs with variable coefficients have been discussed [19], [20], [21], [22]. For example, by using Darboux transformation and generalized Darboux transformation, the multi-soliton solutions, breathers and rogue waves of a sixth-order variable-coefficient nonlinear Schrödinger equation have been obtained in Ref. [23]. Based on the Hirota bilinear method and symbolic computation, the N-soliton solutions, Bäcklund transformation and Lax pair of a generalized fifth-order KdV equation with variable coefficients have been presented in Ref. [24]. In this paper, we consider the (2+1)-dimensional coupled nonlinear partial differential equation with variable coefficients uxt+α1(t)uxuxx+α2(t)uxxxx+α3(t)wx+α4(t)uxy+α5(t)uxx+α6(t)uxxxy+α7(t)(uxuxy+uxxuy)+α8(t)wxxy+α9(t)(uxyw+uywx)+α10(t)wwx+α11(t)wxyy=0,wx=uyy,where αi(t),i=1,2,,11 are real functions of t. There are special cases of Eq. as follows:

Case 1: When α3(t)=α4(t)=α5(t)=α8(t)=α9(t)=α10(t)=α11(t)=0, Eq. reduces to a generalized (2+1)-dimensional variable-coefficient breaking soliton equation [19], uxt+α1(t)uxuxx+α2(t)uxxxx+α6(t)uxxxy+α7(t)(uxuxy+uxxuy)=0,which describes the interaction between the Riemann wave propagating along the y-axis and the long wave propagating along the x-axis. The Painlevé property, Bäcklund transformations, Lax pairs and various exact solutions of Eq. (3) when α1(t)=α2(t)=0, α6(t)=1 and α7(t)=4 have been studied [25], [26], [27], [28].

Case 2: When α1(t)=α2(t)=α3(t)=α5(t)=α8(t)=α9(t)=α10(t)=α11(t)=0, Eq. changes into a generalized (2+1)-dimensional variable-coefficient shallow water wave equation [29], uxt+α4(t)uxy+α6(t)uxxxy+α7(t)(uxuxy+uxxuy)=0,which also describes the interaction between Riemann waves propagating along the y-axis and long waves propagating along the x-axis in a fluid. Bilinear Bäcklund transformation, Lax pair, Darboux covariant Lax pair and soliton solutions of Eq. (4) have been obtained in [30].

Case 3: When α4(t)=α5(t)=α6(t)=α7(t)=α8(t)=α9(t)=α10(t)=α11(t)=0, Eq. turns into a (2+1)-dimensional variable-coefficient potential Kadomtsev–Petviashvili (KP) equation [31], uxt+α1(t)uxuxx+α2(t)uxxxx+α3(t)uyy=0.

Some group-invariant solutions, exact periodic kink-wave solution, periodic soliton and doubly periodic solutions have been derived via Lie symmetry approach, homoclinic test technique and extended homoclinic test technique of Eq. (5) with α1(t)=6, and α2(t)=α3(t)=1 [31], [32].

Compared to these special cases, Eq. as a combined equation can be used to describe more complex and rich physical phenomena. Therefore, it is a very meaningful work to study various exact solutions of Eq. . As far as we know, breather wave solutions, lump solutions and N-soliton solutions of Eq. have not been studied. This paper uses Hirota bilinear method and symbolic computation to explore these different types of analytical solutions of Eq. . The main contents of this paper are arranged as follows:

In Section 2, we construct the Hirota bilinear form of Eq. . In Section 3, the breather wave solutions are obtained by using the extended homoclinic breather technique with a test function. In Section 4, comparing with the lump solutions of Eq. in Ref. [18], we obtain a more general lump solutions of Eq. through the generalized positive quadratic function method. In Section 5, we investigate the N-soliton solutions of Eq. . Especially, the one-soliton and two-soliton solutions are explicitly presented. In Section 6, the conclusion are presented.

Section snippets

Hirota bilinear form

Under the transformation u=u(x,y,t)=2(logf(x,y,t))x,the Eq. becomes fxtffxft+α2(t)(ffxxxx4fxfxxx+3fxx2)+α3(t)(fyyffy2)+α4(t)(fxyffxfy)+α5(t)(fxxffx2)+α6(t)(fxxxyffyfxxx3fxfxxy+3fxxfxy)+α8(t)(fxyyyffxfyyy3fyfxyy+3fyyfxy)+α11(t)(ffyyyy4fyfyyy+3fyy2)=0.Hence we obtain the Hirota bilinear form of Eq. as (DxDt+α2(t)Dx4+α3(t)Dy2+α4(t)DxDy+α5(t)Dx2+α6(t)Dx3Dy+α8(t)Dy3Dx+α11(t)Dy4)ff=0,with the following constraints α1(t)=6α2(t),α7(t)=3α6(t),α9(t)=3α8(t),α10(t)=6α11(t),where D is the Hirota

Breather wave solutions

Breather wave solutions are periodic soliton solutions in space, which is commonly used to explain the generation of rogue waves and the nonlinear stage of modulation instability. In order to find the breather wave solutions of NLPDEs with variable coefficients, based on the Hirota bilinear method, it can be obtained by using the extended homoclinic breather technique [33], [34], [35], [36], [37], the conjugate parameter method [38], and so on. In this paper, we use the extended homoclinic

Lump solutions

Lump solutions are the rational solitons which are local in all directions of space, attenuating and do not have singularities. They appear in many physical phenomena, such as plasma and optical media. On the basis of symbolic computation method, one can use the generalized positive quadratic function [39], [40], [41], [42], [43], [44] to study the lump solutions.

To construct the lump solutions for Eq. , a direct assumption of variable coefficients is taken as follows: f=γ3(t)+(γ1(t)dt+l1x+m1y)

N-soliton solutions

In mathematical physics, N-soliton solutions are very helpful for exploring nonlinear wave phenomena. Breather, complexiton, lump and rogue wave solutions are all special simplifications of N-soliton solutions in different situations. In this section, we use the Hirota bilinear method to investigate N-soliton solutions.

First, we use the form of the function f as follows [45], [46], [47]: fN=σ=0,1exp(i<jNσiσjmij+i=1Nσiθi),where the sums are taken over all possible combinations of σi=0,1,i=1,2,

Conclusion

In this paper, a (2+1)-dimensional coupled nonlinear partial differential equation with variable coefficients is investigated. Based on Hirota’s bilinear form and symbolic computation, the breather wave solutions, lump solutions and N-soliton solutions are presented. By using the extended homoclinic breather technique, the breather wave solutions are constructed, and some lump waves including dark, bright and bright–dark lump waves are also obtained by the generalized positive quadratic

CRediT authorship contribution statement

Qianqian Li: Software, Data curation, Writing – original draft. Wenrui Shan: Conceptualization, Methodology, Writing – review & editing. Panpan Wang: Visualization, Investigation. Haoguang Cui: Supervision, Validation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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