Lie Symmetries and Conservation Laws of Fokas–Lenells Equation and Two Coupled Fokas–Lenells Equations by the Symmetry/Adjoint Symmetry Pair Method

Lihua Zhang,Lingshu Wang,Qianqian Zhao,Gangwei Wang

doi:10.3390/sym14020238

Abstract

The Fokas–Lenells equation and its multi-component coupled forms have attracted the attention of many mathematical physicists. The Fokas–Lenells equation and two coupled Fokas–Lenells equations are investigated from the perspective of Lie symmetries and conservation laws. The three systems have been turned into real multi-component coupled systems by appropriate transformations. By procedures of symmetry analysis, Lie symmetries of the three real systems are obtained. Explicit conservation laws are constructed using the symmetry/adjoint symmetry pair method, which depends on Lie symmetries and adjoint symmetries. The relationships between the multiplier and the adjoint symmetry are investigated.

Highlights

Based on the original work of Noether, one symmetry corresponds to a conservation law for partial differential equations (PDEs) with classical Lagrangians [3]
Taking conservation laws Equation (19) as an example, we show that adjoint symmetries are multipliers corresponding to conservation laws
They can be regarded as integrable analogs of the nonlinear Schrödinger (NLS) equation and its coupled forms in the ultra-short regime

Summary

Introduction

For nonlinear differential equations [1,2,3,4,5,6,7,8,9,10,11,12], differential-difference equations [13,14,15], and fractional differential equations [16,17,18,19,20], it is important to admit Lie symmetries [1,2,10,15]. Lie Symmetries and Conservation Laws of Fokas–Lenells Equation and Two Coupled Fokas–Lenells Equations by the Symmetry/Adjoint Symmetry Pair Method. Based on the obtained Lie symmetries, we can construct conservation laws for Equations (4)–(6) by the steps of the SA method [5,6].

Results

Conclusion