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The Neumann-Moser dynamical system and the Korteweg-de Vries hierarchy

Polina Baron

Abstract

At the focus of the paper are applications of the well-known Moser transformation of the C. Neumann dynamical system. It yields us a new quadratic integrable dynamical system on $\mathbb{C}^{3n+1}$, which we call the Neumann-Moser dynamical system. We present an explicit formula of the inverse of the Moser transformation. Consequently, we obtain explicitly an invertible transformation of the Uhlenbeck-Devaney integrals of the Neumann system into the integrals of our system. One of the main results of the paper is the recurrent solutions of the Neumann-Moser system. We show that every solution of our system solves the Mumford dynamical system, and vice versa. Every solution of the Neumann-Moser system is proven to solve the stationary Korteweg-de Vries hierarchy. As a corollary, we construct explicit solutions of the Neumann-Moser system in hyperelliptic Kleinian functions.

The Neumann-Moser dynamical system and the Korteweg-de Vries hierarchy

Abstract

At the focus of the paper are applications of the well-known Moser transformation of the C. Neumann dynamical system. It yields us a new quadratic integrable dynamical system on , which we call the Neumann-Moser dynamical system. We present an explicit formula of the inverse of the Moser transformation. Consequently, we obtain explicitly an invertible transformation of the Uhlenbeck-Devaney integrals of the Neumann system into the integrals of our system. One of the main results of the paper is the recurrent solutions of the Neumann-Moser system. We show that every solution of our system solves the Mumford dynamical system, and vice versa. Every solution of the Neumann-Moser system is proven to solve the stationary Korteweg-de Vries hierarchy. As a corollary, we construct explicit solutions of the Neumann-Moser system in hyperelliptic Kleinian functions.
Paper Structure (14 sections, 40 theorems, 131 equations)

This paper contains 14 sections, 40 theorems, 131 equations.

Table of Contents

  1. The Neumann dynamical system
  2. The Moser transformation
  3. The Neumann--Moser dynamical system
  4. Integrals
  5. Integrals of the Neumann--Moser system
  6. Integrals of the Neumann system
  7. Transformation of integrals
  8. Recurrent Solutions of the Neumann--Moser system
  9. From the Neumann--Moser system to the Mumford system and back
  10. Connections to the Korteweg--de Vries hierarchy
  11. The Korteweg--de Vries hierarchy
  12. The hyperelliptic Kleinian functions
  13. Solutions in Hyperelliptic Kleinian functions
  14. Matrices in symmetric polynomials

Key Result

Theorem 1

The Moser transformation is invertible up to the sheet of the branched covering. As a corollary, solutions of the Neumann system can be obtained from solutions of the Neumann--Moser system.

Theorems & Definitions (72)

  • Theorem 1: See Lemmas \ref{['Mos-lemma-1']} and \ref{['Mos-lemma-2']}, Corollary \ref{['Moser-cor']}
  • Theorem 2: See Theorem \ref{['F-h_th']}
  • Theorem 3: See Theorem \ref{['u-rec-th']} and Corollary \ref{['u-rec-cor-2']}
  • Theorem 4: See Theorem \ref{['M-NS-th']}
  • Theorem 5: See Theorems \ref{['KdV-NS-th']} and \ref{['KdV-hkf-th']}
  • Definition 1.1
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • ...and 62 more