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

Sergei Igonin

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.

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

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.
Paper Structure (4 sections, 3 theorems, 108 equations)

This paper contains 4 sections, 3 theorems, 108 equations.

Table of Contents

  1. Introduction and the main ideas
  2. Results on matrix Lax representations
  3. Lax representations and Miura-type transformations for integrable equations related to the Narita--Itoh--Bogoyavlensky lattice
  4. Integrable equations, Lax representations, and Miura-type transformations related to the Zhang--Chen lattice

Key Result

Theorem 1

Let $\mathrm{d}\in\mathbb{Z}_{>0}$. Consider an invertible $\mathrm{d}\times\mathrm{d}$ matrix-function $\mathbf{M}=\mathbf{M}(u_0,u_1,u_2,\uplambda)$. Suppose that where $a_0\in\mathbb{C}^N$ is a constant vector and $\mathbf{M}(a_0,u_0,u_1,\uplambda)={\mathcal{S}}^{-1}(\mathbf{M}(a_0,u_1,u_2,\uplambda))$. The matrix $\mathbf{M}(a_0,u_1,u_2,\uplambda)$ is obtained from $\mathbf{M}(u_0,u_1,u_2,\up

Theorems & Definitions (18)

  • Remark 1
  • Definition 1
  • Remark 2
  • Remark 3
  • Definition 2
  • Definition 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • ...and 8 more