Lax Equations, Singularities and Riemann-Hilbert Problems

TL;DR

This paper explores the relationship between singularities in solutions of Lax equations and Toeplitz operator kernels, utilizing factorization on Riemann surfaces to characterize and analyze these singularities.

Contribution

It develops a method linking singularities of Lax equations to Toeplitz operator kernels via Riemann surface factorization, extending classical approaches.

Findings

Singularities correspond to non-trivial Toeplitz kernels.

Riemann surface factorization characterizes singularity locations.

Classical and new methods yield consistent singularity sets.

Abstract

The existence of singularities of the solution for a class of Lax equations is investigated using a development of the fac- torization method first proposed by Semenov-Tian-Shansky and Reymann [11], [9]. It is shown that the existence of a singularity at a point t = ti is directly related to the property that the ker- nel of a certain Toeplitz operator (whose symbol depends on t) be non-trivial. The investigation of this question involves the factor- ization on a Riemann surface of a scalar function closely related to the above-mentioned operator. An example is presented and the set of singularities is shown to coincide with the set obtained by classical methods. This comparison involves relating the two Riemann surfaces associated to the system by these methods.