The Neumann-Moser dynamical system and the Korteweg-de Vries hierarchy
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916141361790976 |
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| author | Baron, Polina |
| author_facet | Baron, Polina |
| contents | At the focus of the paper are applications of the well-known Moser transformation of the C. Neumann dynamical system. It yields us a new quadratic integrable dynamical system on $\mathbb{C}^{3n+1}$, which we call the Neumann-Moser dynamical system. We present an explicit formula of the inverse of the Moser transformation. Consequently, we obtain explicitly an invertible transformation of the Uhlenbeck-Devaney integrals of the Neumann system into the integrals of our system. One of the main results of the paper is the recurrent solutions of the Neumann-Moser system. We show that every solution of our system solves the Mumford dynamical system, and vice versa. Every solution of the Neumann-Moser system is proven to solve the stationary Korteweg-de Vries hierarchy. As a corollary, we construct explicit solutions of the Neumann-Moser system in hyperelliptic Kleinian functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_18079 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Neumann-Moser dynamical system and the Korteweg-de Vries hierarchy Baron, Polina Exactly Solvable and Integrable Systems Mathematical Physics Dynamical Systems 37K10 (Primary), 35Q53, 37J35, 37N05, 70H06 (Secondary) At the focus of the paper are applications of the well-known Moser transformation of the C. Neumann dynamical system. It yields us a new quadratic integrable dynamical system on $\mathbb{C}^{3n+1}$, which we call the Neumann-Moser dynamical system. We present an explicit formula of the inverse of the Moser transformation. Consequently, we obtain explicitly an invertible transformation of the Uhlenbeck-Devaney integrals of the Neumann system into the integrals of our system. One of the main results of the paper is the recurrent solutions of the Neumann-Moser system. We show that every solution of our system solves the Mumford dynamical system, and vice versa. Every solution of the Neumann-Moser system is proven to solve the stationary Korteweg-de Vries hierarchy. As a corollary, we construct explicit solutions of the Neumann-Moser system in hyperelliptic Kleinian functions. |
| title | The Neumann-Moser dynamical system and the Korteweg-de Vries hierarchy |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics Dynamical Systems 37K10 (Primary), 35Q53, 37J35, 37N05, 70H06 (Secondary) |
| url | https://arxiv.org/abs/2402.18079 |