Unique continuation properties for solutions to the Camassa-Holm equation and related models

Abstract:

It is shown that if $u(x,t)$ is a real 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 $u(x,t)=0,(x,t)\in \mathbb {R}\times [0,T]$. 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. This result also applies to solutions of the initial periodic boundary value problems associated to these models.