Integrable Mappings and Soliton Lattices

Abstract

Solitons are found as solutions of integrable partial differential equations (PDE’s) which have been extensively studied during the last three decades. More recently there has been increasing interest in partial difference equations with fields defined at the sites of 2- and 3-dimensional (2D and 3D) lattices. They are the discrete-time analogues of PDE’s as well as of partial difference equations depending on time as well as on 1 or 2 discrete variables. During the last years a variety of integrable 2D and 3D lattices has been found, by the direct linearization method (DLM) which is based on a linear integral equation with arbitrary measure and contour [1]–[4], as well as by the bilinearization method [5] and the τ-function approach [6]. By taking continuum limits the lattice equations yield hierarchies of integrable partial difference equations (or PDE’s) with one or more continuous variables together with a Poisson structure and an infinite number of conserved quantities in involution [7,8], see also ref. [9].KeywordsPoisson BracketPoisson StructureComplete IntegrabilityLinear Integral EquationVertical StepThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.