Consistent Riccati expansion solvability and soliton–cnoidal wave solutions of a coupled KdV system

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Abstract

The consistent Riccati expansion (CRE) solvability of a coupled KdV system is investigated with the aid of the Riccati equation. Differing from the complete Painlevé integrability for three branches, this coupled KdV system is shown to be CRE solvable for one branch and non-CRE solvable for two other branches. Based on the last consistent differential equation under the CRE solvable case, a kind of soliton–cnoidal wave interaction solution is derived explicitly.

Introduction

The study of integrability properties and exact solutions of nonlinear wave systems have always attracted much attention in mathematical physics. A series of effective methods such as inverse scattering transformation, Darboux transformation, Bäcklund transformation, Hirota bilinear method, Painlevé analysis and symmetry reduction have been developed during the past decades. Very recently, a consistent Riccati expansion (CRE)/consistent tanh expansion (CTE) method [1], [2], [3], [4] has been proposed to solve nonlinear wave systems by means of the Riccati equation/tanh function. This expansion method implies a kind of solvability with the consistent overdetermined systems and then this CRE/CTE solvability is demonstrated in various integrable models. It has also been shown that these CRE solvable systems allow the interesting interaction solutions between soliton and other nonlinear waves such as cnoidal wave, Painlevé wave, Airy wave and Bessel wave [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]. In addition, the possible CRE solvable classification is in accordance with that of the Painlevé analysis and the higher-order symmetry consideration [1], [2], [3].

In this paper, we will concentrate on investigating the CRE solvability and the soliton–cnoidal wave interaction solution of the following coupled KdV system [26] ut+uxxx74uuxvvx+54(uv)x=0,vt+vxxx52uux74vvx+2(uv)x=0. This coupled hydrodynamic model may be used to describe the typical resonant interaction of two wave modes in a shallow stratified liquid. Its integrability was verified in the sense of possessing Lax pair and passing Painlevé test, and some special exact solutions were constructed through the Bäcklund transformation [26]. The integrability of the coupled KdV system (1)–(2) was also examined by means of the Hirota’s bilinear method and multiple (singular) soliton solutions were derived formally [27]. Lie symmetry analysis, exact solutions and conservation laws of the coupled KdV system (1)–(2) were obtained in [28].

The rest of this work is organized as follows. In Section 2, the CRE method is performed to the coupled KdV system (1)–(2). As a result, this coupled model is proved to be CRE solvable for one branch and non-CRE solvable for two other branches. This differs from its Painlevé integrability in which the system (1)–(2) was shown to pass the Painlevé test for three branches. In Section 3, starting from the last consistent differential equation in the CRE solvable case, soliton and soliton–cnoidal wave interaction solutions are obtained explicitly. Conclusions are presented in last section.

Section snippets

CRE solvability of the coupled KdV system

In order to compare with the Painlevé integrability of the coupled KdV system (1)–(2), we briefly recall the singularity analysis procedure in Ref. [26]. For the following Laurent expansions u=j=0ujϕj+α,v=j=0vjϕj+β,the singular manifold ϕ and the coefficients uj, vj (j=0,1,2,) are arbitrary functions of x and t. According to the leading order analysis, one can obtain α=β=2 and three possible branches as follows: (i)u0=16ϕx2,v0=32ϕx2;(ii)u0=48ϕx2,v0=48ϕx2;(iii)u0=64ϕx2,v0=80ϕx2.

Soliton–cnoidal wave interaction solutions

Following Theorem 2.1, it can be seen that the solution of the coupled KdV system (1)–(2) can be obtained by solving the associated ϕ Eq. (15). In the following, we choose the tanh function solution of the Riccati Eq. (7) R(ϕ)=12a2a1+δtanhδ2ϕ,to present the exact solutions explicitly. Then the CRE solution (18), (19) is converted to the following CTE solution u=16ϕxxxϕx8ϕxx2ϕx28δϕx2163λ32δϕxxtanhδ2ϕ+16δϕx2tanh2δ2ϕ,v=20ϕxxxϕx10ϕxx2ϕx210δϕx2203λ40δϕxxtanhδ2ϕ+20δϕx2tanh2δ2ϕ. Furthermore,

Conclusions

The CRE solvability and exact interaction solutions of the coupled KdV system are studied through the CRE method. Compared with the complete Painlevé integrability for three branches, it is shown that this coupled KdV system is CRE solvable for one branch and non-CRE solvable for two other branches. For the CRE solvable case, exact soliton and soliton–cnoidal wave interaction solutions are constructed explicitly from the last consistent differential equation.

CRediT authorship contribution statement

Huiling Wu: Investigation, Methodology, Writing – original draft. Junfeng Song: Methodology, Software, Supervision, Writing – review & editing. Quanyong Zhu: Visualization, Formal analysis, Writing – review & editing.

Acknowledgment

This work was supported by the Zhejiang Province Natural Science Foundation of China (Grant No. 2022SJGYZC01).

References (28)

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