Conservation Laws of Discrete Evolution Equations by Symmetries and Adjoint Symmetries

Wen-Xiu Ma

doi:10.3390/sym7020714

Abstract

A direct approach is proposed for constructing conservation laws of discrete evolution equations, regardless of the existence of a Lagrangian. The approach utilizes pairs of symmetries and adjoint symmetries, in which adjoint symmetries make up for the disadvantage of non-Lagrangian structures in presenting a correspondence between symmetries and conservation laws. Applications are made for the construction of conservation laws of the Volterra lattice equation.

Highlights

Noether’s theorem tells us that a symmetry of a differential equation leads to a conservation law of the same equation, if the equation is derived from a Lagrangian, namely, it has a Lagrangian formulation [1,2]
It is easy to see that two local vector fields σ ∈ Aq and ρ ∈ Aq are a symmetry and an adjoint symmetry of the discrete evolution Equation (8), if and only if they satisfy
We have established a direct approach for constructing conservation laws and conserved densities of discrete evolution equations, whether the evolution equations are derivable from a Lagrangian or not

Summary

Introduction

Noether’s theorem tells us that a symmetry of a differential equation leads to a conservation law of the same equation, if the equation is derived from a Lagrangian, namely, it has a Lagrangian formulation [1,2]. The Lagrangian formulation of the equation is essential for presenting conservation laws from symmetries, and many physically important examples can be found in [1,2,3,4]. In this paper, like to show that it is possible to give a positive answer to the above question if we adopt adjoint symmetries. A good attempt to use adjoint symmetries in computing conservation laws of differential equations was made in [5], and the approach utilizes adjoint symmetries, in which an adjoint invariance condition. We will utilize pairs of symmetries and adjoint symmetries to present a direct formula for constructing conservation laws, and conserved densities, for evolution equations.

General Theory

Applications to the Volterra Lattice Equation

Concluding Remarks