Transformations And Reductions In Discrete Integrable Systems

Transformations and reductions in discrete integrable systems are significant in studying the relations between equations and constructing exact solutions.This thesis can be divided into three parts.First,we construct Backlund transformations by decomposition in the ABS list.Each equation in the ABS list admits a beautiful decomposition.We first revisit these decomposition formulas,by which we construct Backlund transformations and consistent triplets.Some BTs are used to construct new solutions,lattice equations and weak Lax pairs.Secondly,we derive rational solutions for the lattice modified Kordeweg-de Vries equation,and Q2,Q1(?),H3*(?),H3(?),H2 and H1 in the ABS list.Backlund transformations between these lattice equations are used.All these rational solu-tions are related to a unified ? function in Casoratian form which obeys a bilinear superposition formula.In addition,1-soliton solutions are also derived by BTs in multi-dimension consistency.Starting from multi-soliton solutions,multi-soliton so-lutions in Casoratian forms are proved by bilinear approach.Thirdly,we study some discrete integrable systems in the frame of bidifferential graded algebras.We construct Lax pairs for relatively general equations in bidiffer-ential graded algebras,and obtain its connections with a certain bicomplex.In this framework,discrete integrable systems including H1 equation are derived.Exact solutions and conservation laws for a typical equation are also constructed.