Abstract

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.

中文翻译：

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芒福德动力系统和超椭圆克莱因函数

摘要

我们发展了芒福德动力系统的微分代数理论。在这个理论的框架中，我们引入了\((P,Q)\)递归，它定义了一个函数序列\(P_1,P_2,\ldots\) ，给定该序列的第一个函数\(P_1\)和参数序列\(h_1,h_2,\dots\)。\((P,Q)\)递归的通解给出了参数分级 Korteweg–de Vries 层次结构的解。我们证明了 Mumford 动力学\(g\)系统的所有解都由条件\(P_{g+1} = 0\)下的\((P,Q)\)递归确定，这等价到函数\(P_1\)的阶数\(2g\)的普通非线性微分方程。明确描述了 Mumford 的\(g\)系统简化为 Buchstaber–Enolskii–Leykin 动力系统，并给出了其在超椭圆 Klein 函数中的显式\(2g\)参数解。