Singularities of Whitham flows for hyperelliptic spectral curves

TL;DR

This paper studies the behavior of Whitham flows on hyperelliptic spectral curves, demonstrating how these flows can be extended through singularities caused by roots of meromorphic differentials.

Contribution

It proves the existence of non-empty stable and unstable manifolds and extends the Whitham flow continuously through singularities.

Findings

Stable and unstable manifolds are non-empty.

Whitham flow can be extended through singularities.

Singularities occur when the differential has roots at fixed points.

Abstract

We consider the Whitham equations for deformations of hyperelliptic spectral curves, which preserve all periods of a meromorphic differential. If the meromorphic differential has a root at a fixed point of the hyperelliptic involution, then the Whitham flow has a singularity. We prove that the stable and unstable manifolds are non-empty and extend the Whitham flow continuously through the singularity.