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The Mumford Dynamical System and Hyperelliptic Kleinian Functions

Victor Buchstaber

Abstract

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 $P_1,P_2,\ldots$ given the first function of this sequence $P_1$ and a sequence of parameters $h_1,h_2,\ldots$. 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.

The Mumford Dynamical System and Hyperelliptic Kleinian Functions

Abstract

We establish differential-algebraic theory of the Mumford dynamical system. In the framework of this theory, we introduce the -recursion, which defines a sequence of functions given the first function of this sequence and a sequence of parameters . The general solution of the -recursion is shown to give a solution for the parametric graded Korteweg--de Vries hierarchy. We prove that all solutions of the Mumford dynamical -system are determined by the -recursion under the condition , which is equivalent to an ordinary nonlinear differential equation of order for the function . Reduction of the -system of Mumford to the Buchstaber--Enolskii--Leykin dynamical system is described explicitly, and its explicit -parameter solution in hyperelliptic Klein functions is presented.
Paper Structure (13 sections, 30 theorems, 72 equations)

This paper contains 13 sections, 30 theorems, 72 equations.

Table of Contents

  1. The Mumford Dynamical System
  2. The Mumford Dynamical System in the Lax Form
  3. The Dynamical System on $\mathbb{C}^{3g+1}$
  4. Integrals of the Mumford Dynamical System
  5. Poisson Brackets on $\mathbb{C}^{3g+1}$
  6. Special Poisson Brackets.
  7. The Vanhaecke brackets.
  8. Hamiltonians of the Mumford System and Casimirs of Brackets $\{\,\boldsymbol\cdot\,,\,\boldsymbol\cdot\,\}_k$.
  9. Differential Polynomials
  10. The $(P,Q)$-recursion
  11. The Gelfand-Dikii recursion
  12. Korteweg--de Vries hierarchy
  13. Hyperelliptic curves

Key Result

Lemma 1.1

Suppose that the mapping $L_\xi$ given by Equation f-1-1 satisfies the system f-1. Then $A_\eta(\mathbf{t}) = .$

Theorems & Definitions (42)

  • Lemma 1.1
  • Corollary 1.2
  • Corollary 2.1
  • Example 2.2
  • Lemma 3.1
  • proof
  • Theorem 3.3
  • Corollary 3.4
  • Theorem 4.1
  • proof
  • ...and 32 more