Discrete integrable equations with three independent variables

Abstract

In this paper, we study nonlinear integrable equations with three independent variables of the following types: Toda-type lattices, semi-discrete lattices, and fully discrete Hirota-Miwa type models. It is shown that integrable equations of all three types admit reductions in the form of Darboux-integrable hyperbolic systems. It is important that the transition from one class to another is carried out by means of discretization (continualization) of the above-mentioned reductions with preservation of characteristic integrals. In other words, at the level of reductions, one can establish some correspondence between the classes of 3D models under consideration. In the context of this correspondence, the authors managed to conduct a comparative analysis of the well-known list of integrable Hirota-Miwa type equations, containing 13 equations. It was established that some equations from this list are related by point changes of variables. As a result, the final list of known integrable Hirota-Miwa type equations was reduced to seven. One equation was obtained by discretizing the list of semi-discrete Toda-type equations using characteristic integrals in this paper, probably it is new. For all seven models, associated linear systems (Lax pairs) are given.