Residual Symmetry, Bäcklund Transformation, and Soliton Solutions of the Higher-Order Broer-Kaup System

Yarong Xia,Ruoxia Yao,Xiangpeng Xin,Wen-Xiu Ma

doi:10.1155/2021/9975303

Abstract

Under investigation in this paper is the higher-order Broer-Kaup(HBK) system, which describes the bidirectional propagation of long waves in shallow water. Via the standard truncated Painlevé expansion method, the residual symmetry of this system is derived. By introducing an appropriate auxiliary-dependent variable, the residual symmetry is successfully localized to Lie point symmetries. Via solving the initial value problems, the finite symmetry transformations are presented. However, the solution which obtained from the residual symmetry is a special group invariant solutions. In order to find more general solution of HBK system, we further generalize the residual symmetry method to the consistent tanh expansion (CTE) method and prove that the HBK system is CTE solvable, then the resonant soliton solutions and interaction solutions among different nonlinear excitations are obtained by the CET method.

Highlights

Since the Lie group theory was proposed by Sophus Lie to study differential equation, the symmetry theory has been widely developed to study nonlinear equation
By developing the truncated Painlevé expansion, Lou introduced the definition of consistent Riccati expansion solvable [26], and this method is greatly valid for constructing both possible new integrable systems and interaction solutions between a soliton and other types of nonlinear excitations
From the coefficients of 1/φ in appendix, we find that u1 = φx, v1 = φxx is just the solution of the symmetry equation for the system (1), where u1, v1 are just the residual of truncated Painlevé expansion (2) when u = u0, v = v0 are the solution of system (1); so based on the definition of residual symmetry, the nonlocal symmetry (8) is called residual symmetry

Summary

Introduction

Since the Lie group theory was proposed by Sophus Lie to study differential equation, the symmetry theory has been widely developed to study nonlinear equation. The residual symmetries, the consistent tanh expansion solvable, and interaction solutions of the higher-order Broer-Kaup system have not yet been studied, which is the prime objective of this paper. One can naturally believe that we need to transform the nonlocal symmetries into local ones To this end, we introduce new variables f and g to eliminate the space derivatives of φ by f = φx, g = f x, ð9Þ the nonlocal symmetry of the higher-order Broer-Kaup system (1) is localized to the following Lie point symmetries σu = f , σv = g, σφ = −φ2, σf = −2φf , σg = −2f 2 − 2φg, ð10Þ for the prolonged systems (1), (7), and (9) with the Lie point symmetry vector. We consider the special case of CRE-consistent Tanh expansion (CTE), which is a more generalized but much simpler method to find interaction solutions between solitons and other nonlinear excitations, such as soliton-resonant solutions, soliton and condial wave, and soliton and sin-cosine wave [38,39,40]

Consistent Tanh Expansion Solvability of the Higher-Order Broer-Kaup System

Explict Solution to the Higher-Order BroerKaup System

Soliton Interactions with Periodic Waves

Summary and Discussion