On integration of the closed KdV dressing chain

TL;DR

This paper integrates the dressing chain equations for second order Sturm-Liouville operators, providing explicit elliptic solutions and analyzing their spectral and bihamiltonian structures.

Contribution

It offers the first explicit elliptic solutions for the closed KdV dressing chain and clarifies the role of integrals in its bihamiltonian structure.

Findings

Explicit elliptic solutions in Weierstrass functions

Connection between closure at third step and genus 1 spectral curve

Dependence of solutions on bihamiltonian integrals

Abstract

The dressing chain equations for the second order Sturm-Liouville differential operators is integrated. The simplest closure at the third step is linked to the spectral curve of the genus 1 and the explicit solution in elliptic Weierstrass functions is given. Dependence on integrals that enter the bihamiltonian structure is specified.