Geodesic
The Mumford dynamical system and hyperelliptic Kleinian functions
Funkcionalʹnyj analiz i ego priloženiâ, Tome 57 (2023) no. 4, pp. 27-45

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We develop a 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 $P_1$ of this sequence and a sequence of parameters $h_1,h_2,\dots$ . 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.
Keywords: Korteweg–de Vries equation, parametric KdV hierarchy, family of Poisson brackets, Gelfand–Dikii recursion, hyperelliptic Kleinian functions. 
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     author = {V. M. Buchstaber},
     title = {The {Mumford} dynamical system and hyperelliptic {Kleinian} functions},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {27--45},
     publisher = {mathdoc},
     volume = {57},
     number = {4},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2023_57_4_a2/}
}
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V. M. Buchstaber. The Mumford dynamical system and hyperelliptic Kleinian functions. Funkcionalʹnyj analiz i ego priloženiâ, Tome 57 (2023) no. 4, pp. 27-45. http://geodesic.mathdoc.fr/item/FAA_2023_57_4_a2/