Unique Continuation Properties for solutions to the Camassa-Holm equation and other non-local equations

TL;DR

This paper proves unique continuation properties for solutions to the Camassa-Holm equation and similar non-local equations, showing that solutions vanishing in an open set must be identically zero, with implications for related models.

Contribution

It establishes unique continuation results for the Camassa-Holm equation and extends the methodology to a broader class of non-local nonlinear equations.

Findings

Solutions vanishing in an open set are identically zero

Results apply to periodic boundary value problems

Method extends to other non-local models like Degasperis-Procesi

Abstract

It is shown that if $\,u(x,t)\,$ is a solution of the initial value problem for the Camassa-Holm equation which vanishes in an open set $\,\Omega\subset \mathbb R\times [0,T]$, then $\,u(x,t)=0,\,(x,t)\in\mathbb R\times [0,T]$. This result also applies to solutions of the initial periodic boundary value problems associated to the Camassa-Holm equation. The argument of proof can be placed in a general setting to extend the above results to a class of non-linear non-local 1-dimensional models which includes the Degasperis-Procesi equation.