A non-commutative discrete first Painlevé hierarchy: the Lax pair approach
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866916059341127680 |
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| author | Bobrova, Irina |
| author_facet | Bobrova, Irina |
| contents | Using a non-commutative analogue of the isomonodromic problem associated with the discrete first Painlevé hierarchy, we construct a non-commutative version of this hierarchy, denoted by $\text{d-PI}_m^{\text{nc}}$. We show that both hierarchies, $\text{d-PI}_m$ and $\text{d-PI}_m^{\text{nc}}$, can be expressed in terms of the polynomials $S_s^k(n)$, which we call the Svinin polynomials. We also derive a reduction of the non-commutative Volterra lattice hierarchy to the $\text{d-PI}_m^{\text{nc}}$ hierarchy and present explicit continuous limits for the first three members of the $\text{d-PI}_m^{\text{nc}}$, thereby recovering non-commutative analogues of the first three members of the differential first Painlevé hierarchy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_29722 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A non-commutative discrete first Painlevé hierarchy: the Lax pair approach Bobrova, Irina Exactly Solvable and Integrable Systems Mathematical Physics Rings and Algebras Primary 46L55. Secondary 39A05, 39A10 Using a non-commutative analogue of the isomonodromic problem associated with the discrete first Painlevé hierarchy, we construct a non-commutative version of this hierarchy, denoted by $\text{d-PI}_m^{\text{nc}}$. We show that both hierarchies, $\text{d-PI}_m$ and $\text{d-PI}_m^{\text{nc}}$, can be expressed in terms of the polynomials $S_s^k(n)$, which we call the Svinin polynomials. We also derive a reduction of the non-commutative Volterra lattice hierarchy to the $\text{d-PI}_m^{\text{nc}}$ hierarchy and present explicit continuous limits for the first three members of the $\text{d-PI}_m^{\text{nc}}$, thereby recovering non-commutative analogues of the first three members of the differential first Painlevé hierarchy. |
| title | A non-commutative discrete first Painlevé hierarchy: the Lax pair approach |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics Rings and Algebras Primary 46L55. Secondary 39A05, 39A10 |
| url | https://arxiv.org/abs/2605.29722 |