On matrix Lax representations for (1+1)-dimensional evolutionary differential-difference equations
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| Format: | Preprint |
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2026
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| _version_ | 1866915967810928640 |
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| author | Igonin, Sergei |
| author_facet | Igonin, Sergei |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_26870 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On matrix Lax representations for (1+1)-dimensional evolutionary differential-difference equations Igonin, Sergei Exactly Solvable and Integrable Systems Mathematical Physics Dynamical Systems 37K60, 37K35 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. |
| title | On matrix Lax representations for (1+1)-dimensional evolutionary differential-difference equations |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics Dynamical Systems 37K60, 37K35 |
| url | https://arxiv.org/abs/2604.26870 |