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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.04472 |
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| _version_ | 1866911041924890624 |
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| author | Xenitidis, Pavlos |
| author_facet | Xenitidis, Pavlos |
| contents | Employing the Lax pairs of the noncommutative discrete potential Korteweg--de Vries (KdV) and Hirota's KdV equations, we derive differential--difference equations that are consistent with these systems and serve as their generalised symmetries. Miura transformations mapping these equations to a noncommutative modified Volterra equation and its master symmetry are constructed. We demonstrate the use of these symmetries to reduce the potential KdV equation, leading to a noncommutative discrete Painlev{è} equation and to a system of partial differential equations that generalises the Ernst equation and the Neugebauer--Kramer involution. Additionally, we present a Darboux transformation and an auto-Bäcklund transformation for the Hirota KdV equation, and establish their connection with the noncommutative Yang--Baxter map $F_{III}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04472 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Noncommutative discrete equations, symmetries and reductions Xenitidis, Pavlos Exactly Solvable and Integrable Systems Mathematical Physics Employing the Lax pairs of the noncommutative discrete potential Korteweg--de Vries (KdV) and Hirota's KdV equations, we derive differential--difference equations that are consistent with these systems and serve as their generalised symmetries. Miura transformations mapping these equations to a noncommutative modified Volterra equation and its master symmetry are constructed. We demonstrate the use of these symmetries to reduce the potential KdV equation, leading to a noncommutative discrete Painlev{è} equation and to a system of partial differential equations that generalises the Ernst equation and the Neugebauer--Kramer involution. Additionally, we present a Darboux transformation and an auto-Bäcklund transformation for the Hirota KdV equation, and establish their connection with the noncommutative Yang--Baxter map $F_{III}$. |
| title | Noncommutative discrete equations, symmetries and reductions |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics |
| url | https://arxiv.org/abs/2507.04472 |