The Mumford Dynamical System and Hyperelliptic Kleinian Functions

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.