Simplifications of Lax pairs for differential–difference equations by gauge transformations and (doubly) modified integrable equations

Abstract

Matrix differential–difference Lax pairs play an essential role in the theory of integrable nonlinear differential–difference equations. We present sufficient conditions which allow one to simplify such a Lax pair by matrix gauge transformations. Furthermore, we describe a procedure for such a simplification and present applications of it to constructing new integrable equations connected by (non-invertible) discrete substitutions of Miura type to known equations with Lax pairs.Suppose that one has three (possibly multicomponent) equations E, E1, E2, a (Miura-type) discrete substitution from E1 to E, and a discrete substitution from E2 to E1. Then E1 and E2 can be called a modified version of E and a doubly modified version of E, respectively. We demonstrate how the above-mentioned procedure helps (in the considered examples) to construct modified and doubly modified versions of a given equation possessing a Lax pair satisfying certain conditions.The considered examples include scalar equations of Itoh–Narita–Bogoyavlensky type and 2-component equations related to the Toda lattice. We present several new integrable equations connected by new discrete substitutions of Miura type to known equations.

Keywords

• Integrable differential–difference equations
• Matrix Lax pairs
• Gauge transformations
• Discrete substitutions of Miura-type
• Itoh–Narita–Bogoyavlensky type equations
• Toda lattice