Residual Symmetries and Bäcklund Transformations of Strongly Coupled Boussinesq–Burgers System

Haifeng Wang,Yufeng Zhang

doi:10.3390/sym11111365

Abstract

In this article, we construct a new strongly coupled Boussinesq–Burgers system taking values in a commutative subalgebra Z 2 . A residual symmetry of the strongly coupled Boussinesq–Burgers system is achieved by a given truncated Painlevé expansion. The residue symmetry with respect to the singularity manifold is a nonlocal symmetry. Then, we introduce a suitable enlarged system to localize the nonlocal residual symmetry. In addition, a Bäcklund transformation is obtained with the help of Lie’s first theorem. Further, the linear superposition of multiple residual symmetries is localized to a Lie point symmetry, and a N-th Bäcklund transformation is also obtained.

Highlights

Integrable equations have wide applications in the field of nonlinear science, such as plasma physics [1,2,3,4], hydrodynamics [5,6,7], nonlinear optics [8,9], and so on [10,11,12,13,14,15]
In [18,19,20], the authors proposed a residual symmetry in the process of the residue of the truncated Painlevé expansion for the bosonized super symmetric KdV equation which is a nonlocal symmetry
In [28,29], the authors were concerned with the application of the nonlocal residual symmetry analysis to the Boussinesq–Burgers (B–B) system, which has the form as follows: pt − ( β − 1) p xx − 2pp x − r x = 0, β rt + ( β − 1)r xx + β( − 1) p xxx − 2( pr ) x = 0, (1a) where v = v( x, t) is the height deviating from the equilibrium position of water, u = u( x, t) is the field of horizontal velocity, β is a constant representing different dispersive power

Summary

Introduction

Integrable equations have wide applications in the field of nonlinear science, such as plasma physics [1,2,3,4], hydrodynamics [5,6,7], nonlinear optics [8,9], and so on [10,11,12,13,14,15]. In [28,29], the authors were concerned with the application of the nonlocal residual symmetry analysis to the Boussinesq–Burgers (B–B) system, which has the form as follows: pt − ( β − 1) p xx − 2pp x − r x = 0, β rt + ( β − 1)r xx + β( − 1) p xxx − 2( pr ) x = 0,. We apply the nonlocal residual symmetry analysis to the strongly coupled Boussinesq–Burgers system. By balancing the dispersion and nonlinear terms according to the leader order analysis to the system Equation (2), the truncated Painlevé expansion has the following form: p0 ψ − q0 φ. We will give the Bäcklund symmetry theorem, which is obtained by using a finite transformation of the Lie point symmetry Equation (10). Bäcklund Transformations of Strongly Coupled Burgers System Related to Multiple

Residual Symmetries

Conclusions