Residual Symmetry Reduction and Consistent Riccati Expansion to a Nonlinear Evolution Equation

Lamine Thiam,Xi-Zhong Liu

doi:10.1155/2019/6503564

Abstract

The residual symmetry of a (1 + 1)-dimensional nonlinear evolution equation (NLEE) ut+uxxx−6u2ux+6λux=0 is obtained through Painlevé expansion. By introducing a new dependent variable, the residual symmetry is localized into Lie point symmetry in an enlarged system, and the related symmetry reduction solutions are obtained using the standard Lie symmetry method. Furthermore, the (1 + 1)-dimensional NLEE equation is proved to be integrable in the sense of having a consistent Riccati expansion (CRE), and some new Bäcklund transformations (BTs) are given. In addition, some explicitly expressed solutions including interaction solutions between soliton and cnoidal waves are derived from these BTs.

Highlights

In nonlinear science, the study of nonlinear equations plays an important role in analyzing related complex phenomenon, which exists in the elds of uid dynamics, plasma, optics, and so on [1]
To concur this di culty, Cheng et al [8] proposed localizing a nonlocal symmetry in an enlarged nonlinear system by introducing some new dependent variables to the original system, provided all the Lie point symmetries are closed in the enlarged system
A lot of works have been done on many important nonlinear systems [9,10,11,12,13]. ey prove that this localization method is very e cient in obtaining new Backlund transformations (BTs) and new symmetry reduction solutions related to a nonlocal symmetry in an enlarged system

Summary

Introduction

The study of nonlinear equations plays an important role in analyzing related complex phenomenon, which exists in the elds of uid dynamics, plasma, optics, and so on [1]. Ey prove that this localization method is very e cient in obtaining new Backlund transformations (BTs) and new symmetry reduction solutions related to a nonlocal symmetry in an enlarged system. On this basis, the finite transformation related to the residual symmetry is obtained by applying Lie’s first theorem. By applying the CTE method, some concrete explicitly expressed solutions of the NLEE are given, which include the interaction solutions between solitons and background cnoidal waves. e last section contains a summary

Localization of Residual Symmetry

CRE Solvability and Interaction Solutions

Conclusion