A note on critical points of integrals of soliton equations

TL;DR

This paper investigates the critical points of integrals of motion for soliton equations within the space of finite-gap solutions, revealing explicit descriptions via real-normalized differentials on spectral curves.

Contribution

It extends the understanding of integrals of motion to the moduli space of Riemann surfaces with marked points, providing explicit characterizations of critical points.

Findings

Critical points characterized by real-normalized differentials

Explicit description of solutions to the variational problem

Connections to problems in mathematical physics

Abstract

We consider the problem of extending the integrals of motion of soliton equations to the space of all finite-gap solutions. We consider the critical points of these integrals on the moduli space of Riemann surfaces with marked points and jets of local coordinates. We show that the solutions of the corresponding variational problem have an explicit description in terms of real-normalized differentials on the spectral curve. Such conditions have previously appeared in a number of problems of mathematical physics.