On the isospectral problem of the dispersionless Camassa-Holm equation

TL;DR

This paper develops spectral theory for the dispersionless Camassa-Holm equation with signed measures, proving uniqueness of spectral data and solving the inverse problem for sign definite measures, with applications to initial condition behavior.

Contribution

It introduces a novel spectral analysis framework allowing signed measures and solves the inverse problem for sign definite measures, advancing understanding of the equation's solutions.

Findings

Weight is uniquely determined by spectral data.

Inverse spectral problem solved for sign definite measures.

Initial conditions split into peakons asymptotically.

Abstract

We discuss direct and inverse spectral theory for the isospectral problem of the dispersionless Camassa--Holm equation, where the weight is allowed to be a finite signed measure. In particular, we prove that this weight is uniquely determined by the spectral data and solve the inverse spectral problem for the class of measures which are sign definite. The results are applied to deduce several facts for the dispersionless Camassa--Holm equation. In particular, we show that initial conditions with integrable momentum asymptotically split into a sum of peakons as conjectured by McKean.