The inverse spectral transform for the conservative Camassa-Holm flow with decaying initial data

TL;DR

This paper develops an inverse spectral transform method for the conservative Camassa-Holm flow with decaying initial data, enabling the proof of weak solutions for the associated Cauchy problem.

Contribution

It introduces a novel inverse spectral transform approach specifically for the Camassa-Holm flow with decay conditions, advancing the analysis of its solutions.

Findings

Established the inverse spectral transform for the flow.

Proved existence of weak solutions for the Cauchy problem.

Extended the spectral analysis to decaying initial data cases.

Abstract

We establish the inverse spectral transform for the conservative Camassa-Holm flow with decaying initial data. In particular, it is employed to prove existence of weak solutions for the corresponding Cauchy problem.