An Approach to Master Symmetries of Lattice Equations

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 [1] [2]. 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 [3] [4]. Moreover these symmetries form beautiful algebraic structures [3] [4]. The appearance of master symmetries [5] 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 [6] and constitute centreless Virasoro algebras together with time-independent symmetries [7]. Moreover this kind of τ -symmetries may be generated by use of Lax equations [8] or zero curvature equations [9]. In the case of lattice equations, there also exist some similar results. For instance, a lot of lattice equations have τ -symmetries and centreless Virasoro symmetry algebras [10]. So far, however, there has not been a systematic theory to construct this kind of τ -symmetries for lattice equations.