An Unusual Series of Autonomous Discrete Integrable Equations on a Square Lattice

Abstract

We present an infinite series of autonomous discrete equations on a square lattice with hierarchies of autonomous generalized symmetries and conservation laws in both directions. Their orders in both directions are equal to κN, where κ is an arbitrary natural number and N is the equation number in the series. Such a structure of hierarchies is new for discrete equations in the case N > 2. The symmetries and conservation laws are constructed using the master symmetries, which are found directly together with generalized symmetries. Such a construction scheme is apparently new in the case of conservation laws. Another new point is that in one of the directions, we introduce the master symmetry time into the coefficients of the discrete equations. In the most interesting case N = 2, we show that a second-order generalized symmetry is closely related to a relativistic Toda-type integrable equation. As far as we know, this property is very rare in the case of autonomous discrete equations.