This article is cited in 44 papers

Lie symmetry, nonlocal symmetry analysis, and interaction of solutions of a $(2+1)$-dimensional KdV–mKdV equation

Zhonglong Zhao^a, Lingchao He<sup>b

a</sup> Department of Mathematics, North University of China, Taiyuan, Shanxi, China
^b College of Mechanical and Vehicle Engineering, Taiyuan University of Technology, Taiyuan, Shanxi, China

Abstract: We use the method of Lie symmetry analysis to investigate the properties of a $(2+1)$-dimensional KdV–mKdV equation. Using the Ibragimov method, which relies only on the existence of the commutator table, we construct an optimal system of one-dimensional subalgebras of the Lie algebra and study invariant solutions and similarity reductions by considering representatives of the optimal system. To analyze some nonlocal symmetry properties, we apply the truncated Painlevé expansion method and obtain two Bäcklund transformations that are not autotransformations and one auto-Bäcklund transformation. To localize the nonlocal symmetry and obtain a local Lie point symmetry, we introduce an expanded system. Using solutions of the corresponding Cauchy problems for Lie point symmetries, we prove a theorem on a finite symmetry transformation and find the $n$th Bäcklund transformation in terms of determinants. Based on one of the obtained Bäcklund transformations that are not autotransformations, we derive lump-type solutions. In addition, we prove the integrability of the equation by the consistent Riccati expansion method. We present explicit soliton-cnoidal wave solutions and investigate the dynamical characteristics of the obtained solutions using numerical analysis.

Keywords: $(2+1)$-dimensional KdV–mKdV equation, Lie point symmetry, nonlocal symmetry, Bäcklund transformation.

PACS: 02.20.-a, 02.30.Jr, 05.45.Yv

Received: 12.09.2020
Revised: 18.10.2020

DOI: 10.4213/tmf9982

 English version: Theoretical and Mathematical Physics, 2021, 206:2, 142–162

Bibliographic databases:

• 
• 
•