Matrix Lax pairs under the gauge equivalence relation induced by the gauge group action and Miura-type transformations for lattice equations
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908713496870912 |
|---|---|
| author | Igonin, Sergei |
| author_facet | Igonin, Sergei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_08579 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Matrix Lax pairs under the gauge equivalence relation induced by the gauge group action and Miura-type transformations for lattice equations Igonin, Sergei Exactly Solvable and Integrable Systems 37K60, 37K35 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. |
| title | Matrix Lax pairs under the gauge equivalence relation induced by the gauge group action and Miura-type transformations for lattice equations |
| topic | Exactly Solvable and Integrable Systems 37K60, 37K35 |
| url | https://arxiv.org/abs/2405.08579 |