Discrete integrable systems: Multidimensional consistency

Abstract

In contrast to the well-established theory of differential equations, the theory of difference equations has not quite developed so far. The most recent advances in the theory of discrete integrable systems have brought a true revolution to the study of difference equations. Multidimensional consistency is a new concept appearing in the research of discrete integrable systems. This property, as an explanation to a type of discrete integrability, plays an important role in constructing the Bäcklund transformations, Lax pairs and exact solutions for discrete integrable system. In the present paper, the multidimensional consistency and its applications in the research of discrete integrable systems are reviewed.