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arXiv · 2405.08579

Matrix Lax pairs under the gauge equivalence relation induced by the gauge group action and Miura-type transformations for lattice equations

Sergei Igonin
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Abstract

In this paper we explore interconnections of differential-difference matrix Lax representations (Lax pairs), gauge transformations, and discrete Miura-type transformations (MTs), which belong to the main tools in the theory of integrable differential-difference (lattice) 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 means of a (local) matrix gauge transformation. Matrix gauge transformations constitute an infinite-dimensional group called the matrix gauge group, which acts naturally on the set of MLRs of a given equation. Two MLRs are gauge equivalent if and only if they belong to the same orbit of the matrix gauge group action. For a wide class of MLRs of (vector) evolutionary differential-difference equations, we present results on the following questions: 1. When and how can one simplify a given MLR by matrix gauge transformations and bring the MLR to a form suitable for constructing MTs? 2. A MLR is called fake if it is gauge equivalent to a trivial MLR. How to determine whether a given MLR is not fake? This allows us to construct new integrable equations (with new MLRs) connected by new MTs to known equations.

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Sergei Igonin. 2024-05-14. Matrix Lax pairs under the gauge equivalence relation induced by the gauge group action and Miura-type transformations for lattice equations. https://doi.org/10.1016/j.geomphys.2025.105585

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