The Mumford Dynamical System and Hyperelliptic Kleinian Functions

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 P₁, P₂, given the first function of this sequence P₁ and a sequence of parameters h₁, h₂,. 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 P₆+₁ = 0, which is equivalent to an ordinary nonlinear differential equation of order 2g for the function P₁. 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.