A (2+1)-dimensional sine-Gordon and sinh-Gordon equations with symmetries and kink wave solutions

Gangwei Wang,Kaitong Yang,Haicheng Gu,Fei Guan,A.H Kara

doi:10.1016/j.nuclphysb.2020.114956

Abstract

In this paper, a (2+1)-dimensional sine-Gordon equation and a sinh-Gordon equation are derived from the well-known AKNS system. Based on the Hirota bilinear method and Lie symmetry analysis, kink wave solutions and traveling wave solutions of the (2+1)-dimensional sine-Gordon equation are constructed. The traveling wave solutions of the (2+1)-dimensional sinh-Gordon equation can also be provided in a similar manner. Meanwhile, conservation laws are derived.

Highlights

It is well-known that the classical (1 + 1)-dimensional sine-Gordon equation utt = uxx + sin u (1)or equivalent formG
Gu and Hu [5] provided explicit solutions to the intrinsic generalization for the wave and sine-Gordon equations, and Hu [7] investigated the relationship between soliton and differential Geometry through the sG equation
Derivation of (2+1)-dimensional sine-Gordon and sinh-Gordon equations. It is well-known that the AKNS system [1] is one of the classical well-known integrable systems from which a great many of nonlinear evolution equations can be derived, such as the famous KdV equation, the MKdV equation, the nonlinear Schrödinger equation (NLS), the Burgers equation, the (1+1)-dimensional sine-Gordon equation, etc

Summary

Introduction

In [25], the authors studied symmetry groups of the intrinsic generalized wave and sine-Gordon equations. Lie symmetries approach is employed to reduce the (2+1)-dimensional sine-Gordon and sinh-Gordon equations so that their traveling wave solutions are obtained. It is well-known that the AKNS system [1] is one of the classical well-known integrable systems from which a great many of nonlinear evolution equations can be derived, such as the famous KdV equation, the MKdV equation, the nonlinear Schrödinger equation (NLS), the Burgers equation, the (1+1)-dimensional sine-Gordon equation, etc. Subsequently yields the following (2+1)-dimensional sine-Gordon equation uxx − uxy − uxt + uyt = sin u.

Multi-kink wave solutions

Determinant representation of the N -kink wave solution

Interaction of kink waves

Conservation laws

Conclusions