An approach to master symmetries of lattice equations

Summary

Abstract

An approach to master symmetries of lattice equations is proposed by the use of discrete zero curvature equation. Its key is to generate isospectral flows from the discrete spectral problem associated with a given lattice equation. A Volterra-type lattice hierarchy and the Toda lattice hierarchy are analyzed as two illustrative examples.

Introduction

Symmetries are one of important aspects of soliton theory. When any integrable character hasn't been found for a given equation, among the most efficient ways is to consider its symmetries. It is through symmetries that Russian scientists et al. classified many integrable equations including lattice equations. They gave some specific description for the integrability of nonlinear equations in terms of symmetries, and showed that if an equation possesses higher differential-difference degree symmetries, then it is subject to certain conditions, for example, the degree of its nonlinearity mustn't be too large, compared with its differential-difference degree. Usually an integrable equation in soliton theory is referred as to an equation possessing infinitely many symmetries. Moreover these symmetries form beautiful algebraic structures.

The appearance of master symmetries gives a common character for integrable differential equations both in 1 + 1 dimensions and in 1 + 2 dimensions, for example, the KdV equation and the KP equation. The resulting symmetries are sometimes called τ-symmetries and constitute centreless Virasoro algebras together with time-independent symmetries.