Local well-posedness of the Camassa-Holm equation on the real line

TL;DR

This paper establishes the local well-posedness of the Camassa-Holm equation on the real line using a Lagrangian approach, proving existence, uniqueness, and continuous dependence of solutions in a specific function space.

Contribution

It extends the local well-posedness results of the Camassa-Holm equation from periodic domains to the real line in a diffeomorphism space.

Findings

Proves local existence and uniqueness of solutions.

Shows continuous dependence on initial data.

Uses Lagrangian formulation and ODE techniques.

Abstract

In this paper we prove the local well-posedness of the Camassa-Holm equation on the real line in the space of continuously differentiable diffeomorphisms with an appropriate decaying condition. This work was motivated by G. Misiolek who proved the same result for the Camassa-Holm equation on the periodic domain. We use the Lagrangian approach and rewrite the equation as an ODE on the Banach space. Then by using the standard ODE technique, we prove existence and uniqueness. Finally, we show the continuous depdendence of the solution on the initial data by using the topological group property of the diffeomorphism group.