The gauge action on semi-discrete Lax representations and its invariants

TL;DR

This paper studies gauge invariants of matrix Lax representations for semi-discrete equations, providing criteria to determine when spectral parameters can be eliminated via gauge transformations, thus aiding integrability analysis.

Contribution

It introduces explicit gauge invariants for semi-discrete Lax pairs and establishes conditions for the removability of spectral parameters, advancing the understanding of integrability in differential-difference equations.

Findings

Invariants depend nontrivially on the spectral parameter if it cannot be gauged away.

Necessary conditions for gauge equivalence of two Lax representations are derived.

Comparison with continuous zero-curvature representations highlights similarities and differences.

Abstract

Semi-discrete (differential-difference) matrix Lax representations (Lax pairs) play an essential role in the theory of integrable differential-difference equations. Fix a (1+1)-dimensional evolutionary differential-difference (semi-discrete) equation and consider matrix Lax representations (MLRs) of this equation. Two MLRs are said to be gauge equivalent if one of them can be obtained from the other by applying a (local) matrix gauge transformation. Gauge transformations (GTs) form an infinite-dimensional group, which acts on the set of MLRs of a given equation. Two MLRs are gauge equivalent iff they belong to the same orbit of this action. When one tries to establish integrability (in the sense of soliton theory) for a given equation, one is interested in MLRs which depend on a parameter (usually called the spectral parameter) such that the parameter cannot be removed by any GT. We introduce and study explicit invariants with respect to the action of GTs on the set of MLRs for a given (1+1)-dimensional evolutionary differential-difference equation with any number of components. Using these invariants, we obtain the following results: - Consider a MLR with a parameter $\lambda$. If at least one of the invariants computed for this MLR depends nontrivially on $\lambda$, then the parameter cannot be removed by any GT. - When we have two different MLRs for a given equation, we present necessary conditions for these two MLRs to be gauge equivalent. Our results on semi-discrete MLRs of differential-difference equations are inspired by results of S$.$Yu. Sakovich and M. Marvan on (continuous) zero-curvature representations of partial differential equations. A comparison with some of the results of S$.$Yu. Sakovich and M. Marvan is presented.