Korteweg-de Vries hierarchy and related completely integrable systems: I. Algebro-geometrical approach

TL;DR

This paper explores the algebro-geometrical structure of the Korteweg-de Vries hierarchy, providing methods for elliptic solutions and deriving matrix Lax representations for related integrable systems.

Contribution

It introduces a general approach for finding elliptic solutions to the KdV hierarchy and related systems, expressing solutions via Novikov polynomials and deriving new Lax representations.

Findings

Elliptic solutions expressed through Novikov polynomials.

Connection of solutions with Hermite and Lamé polynomials.

Derived Lax pairs for several integrable systems.

Abstract

We consider complementary dynamical systems related to stationary Korteweg-de Vries hierarchy of equations. A general approach for finding elliptic solutions is given. The solutions are expressed in terms of Novikov polynomials in general quais-periodic case. For periodic case these polynomials coincide with Hermite and Lam\'e polynomials. As byproduct we derive $2\times 2$ matrix Lax representation for Rosochatius-Wojciechiwski, Rosochatius, second flow of stationary nonlinear vectro Schr\"{o}dinger equations and complex Neumann system.