The Mumford Dynamical System and Hyperelliptic Kleinian Functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917590221193216 |
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| author | Buchstaber, Victor |
| author_facet | Buchstaber, Victor |
| contents | We establish differential-algebraic theory of the Mumford dynamical system. In the framework of this theory, we introduce the $(P,Q)$-recursion, which defines a sequence of functions $P_1,P_2,\ldots$ given the first function of this sequence $P_1$ and a sequence of parameters $h_1,h_2,\ldots$. The general solution of the $(P,Q)$-recursion is shown to give a solution for the parametric graded Korteweg--de Vries hierarchy. We prove that all solutions of the Mumford dynamical $g$-system are determined by the $(P,Q)$-recursion under the condition $P_{g+1} = 0$, which is equivalent to an ordinary nonlinear differential equation of order $2g$ for the function $P_1$. Reduction of the $g$-system of Mumford to the Buchstaber--Enolskii--Leykin dynamical system is described explicitly, and its explicit $2g$-parameter solution in hyperelliptic Klein functions is presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09218 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Mumford Dynamical System and Hyperelliptic Kleinian Functions Buchstaber, Victor Exactly Solvable and Integrable Systems Dynamical Systems 35Q51 We establish differential-algebraic theory of the Mumford dynamical system. In the framework of this theory, we introduce the $(P,Q)$-recursion, which defines a sequence of functions $P_1,P_2,\ldots$ given the first function of this sequence $P_1$ and a sequence of parameters $h_1,h_2,\ldots$. The general solution of the $(P,Q)$-recursion is shown to give a solution for the parametric graded Korteweg--de Vries hierarchy. We prove that all solutions of the Mumford dynamical $g$-system are determined by the $(P,Q)$-recursion under the condition $P_{g+1} = 0$, which is equivalent to an ordinary nonlinear differential equation of order $2g$ for the function $P_1$. Reduction of the $g$-system of Mumford to the Buchstaber--Enolskii--Leykin dynamical system is described explicitly, and its explicit $2g$-parameter solution in hyperelliptic Klein functions is presented. |
| title | The Mumford Dynamical System and Hyperelliptic Kleinian Functions |
| topic | Exactly Solvable and Integrable Systems Dynamical Systems 35Q51 |
| url | https://arxiv.org/abs/2402.09218 |