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.
@article{FAA_2023_57_4_a2,
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/}
}
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/