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The Mumford dynamical system and the Gelfand-Dikii recursion

Polina Baron

Abstract

In his paper "The Mumford Dynamical System and Hyperelliptic Kleinian Functions" (see arXiv:2402.09218), Victor Buchstaber developed the differential-algebraic theory of the Mumford dynamical system. The key object of this theory is the (P,Q)-recursion introduced in his paper. In the present paper, we further develop the theory of (P,Q)-recursion and describe its connections to the Korteweg-de Vries hierarchy, the Lenard operator, and the Gelfand-Dikii recursion.

The Mumford dynamical system and the Gelfand-Dikii recursion

Abstract

In his paper "The Mumford Dynamical System and Hyperelliptic Kleinian Functions" (see arXiv:2402.09218), Victor Buchstaber developed the differential-algebraic theory of the Mumford dynamical system. The key object of this theory is the (P,Q)-recursion introduced in his paper. In the present paper, we further develop the theory of (P,Q)-recursion and describe its connections to the Korteweg-de Vries hierarchy, the Lenard operator, and the Gelfand-Dikii recursion.
Paper Structure (5 sections, 10 theorems, 49 equations)

This paper contains 5 sections, 10 theorems, 49 equations.

Table of Contents

  1. The $(P,Q)$-recursion and its transformation into the $B(u;h)$-recursion
  2. Basic properties of the $B(u;h)$-recursion
  3. Differential form of the $B(u,h)$-recursion and the Lenard operator
  4. Solving the $B(u,h)$-recursion in terms of the $\mathbf{B}(u)$-recursion
  5. Solving the Gelfand-Dikii recursion in terms of the $B(u,h)$-recursion

Key Result

Theorem 2.2

For all $k\geq 0$, the $B(u;h)$-recursion satisfies the equation

Theorems & Definitions (21)

  • Definition 1.1: Buchstaber Buch23
  • Remark 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • ...and 11 more