On matrix Lax representations for (1+1)-dimensional evolutionary differential-difference equations

TL;DR

This paper develops new methods for simplifying and classifying matrix Lax representations in (1+1)-dimensional differential-difference equations, introducing criteria to identify non-trivial cases and constructing new integrable equations.

Contribution

It extends the theory of matrix Lax representations and gauge transformations, providing new tools for constructing and analyzing integrable differential-difference equations.

Findings

New criteria for simplifying matrix Lax representations using gauge transformations.

Construction of several new two-component integrable equations.

Identification of non-fake matrix Lax representations for complex equations.

Abstract

Differential-difference matrix Lax representations (Lax pairs), gauge transformations, and discrete Miura-type transformations (MTs) belong to the main tools in the theory of (nonlinear) integrable differential-difference equations. For a given equation, two matrix Lax representations (MLRs) are said to be gauge equivalent if one of them can be obtained from the other by applying a matrix gauge transformation. Generalizing and extending several previous works on MLRs and MTs, we present new results on the following problems: - When and how can one simplify a given MLR by means of gauge transformations? - How can one use MLRs and gauge transformations for constructing MTs? - A MLR is called fake if it is gauge equivalent to a trivial MLR. How to determine whether a given MLR is not fake? We consider the general (1+1)-dimensional evolutionary differential-difference case when a MLR can depend on any shifts of dependent variables and can be non-autonomous. As applications and illustrations of the presented general theory, we construct several new two-component integrable equations (with new MLRs) connected by new MTs to known integrable equations from the papers [S. Konstantinou-Rizos, A.V. Mikhailov, P. Xenitidis, J. Math. Phys. 2015], [E. Mansfield, G. Mari Beffa, Jing Ping Wang, Found. Comput. Math. 2013]), including non-autonomous examples.