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,… given the first function P_1 of this sequence and a sequence of parameters h_1,h_2,… . 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.