Uniqueness of Conservative Solutions to the Camassa-Holm Equation via Characteristics

TL;DR

This paper proves the uniqueness of conservative solutions to the Camassa-Holm equation by analyzing characteristic curves and their evolution, ensuring a unique, global solution for initial data in H^1.

Contribution

It introduces a new method based on characteristics to establish the uniqueness of solutions to the Camassa-Holm equation.

Findings

Unique characteristic curves through each initial point.

Global in time existence of solutions for initial data in H^1.

Proof of solution uniqueness using characteristic analysis.

Abstract

The paper provides a direct proof the uniqueness of solutions to the Camassa-Holm equation, based on characteristics. Given a conservative solution $u=u(t,x)$, an equation is introduced which singles out a unique characteristic curve through each initial point. By studying the evolution of the quantities $u$ and $v= 2\arctan u_x$ along each characteristic, it is proved that the Cauchy problem with general initial data $u_0\in H^1(\mathbb{R})$ has a unique solution, globally in time.