Singular spectral curves in finite gap integration

TL;DR

This paper explores applications of finite gap integration involving singular spectral curves, including coordinate systems, Frobenius manifolds, and soliton deformations, advancing the understanding of integrable systems with singular spectral data.

Contribution

It introduces novel applications of finite gap integration to singular spectral curves, expanding the scope of integrable systems analysis.

Findings

Spectral curves with geometrical genus zero are used for orthogonal curvilinear coordinates.

Finite gap Frobenius manifolds are constructed using singular spectral curves.

Soliton deformations of spectral curves relate to equations with self-consistent sources.

Abstract

We expose some applications of the finite gap integration method which involve singular spectral curves. They include orthogonal curvilinear coordinate systems corresponding to spectral curves with geometrical genus zero, finite gap Frobenius manifolds, and soliton deformations of spectral curves corresponding to equations with self-consistent sources. This is an extended version of the talk given at the conference "Geometry, Dynamics, Integrable Systems --- GDIS 2010" (Serbia, September 7--13, 2010).