On special properties of solutions to Camassa-Holm equation and related models

TL;DR

This paper investigates the unique continuation properties of solutions to the Camassa-Holm and related equations, demonstrating the optimality of previous results and clarifying conditions under which solutions can be uniquely continued.

Contribution

It establishes the optimality of known unique continuation results for the Camassa-Holm and Degasperi-Procesi models, especially concerning the role of the constant parameter.

Findings

Unique continuation results are optimal for the case c_0=0.

Results fail for any constant c_0≠ 0.

The study applies to both initial value and periodic boundary value problems.

Abstract

We study unique continuation properties of solutions to the b-family of equations. This includes the Camassa-Holm and the Degasperi-Procesi models. We prove that for both, the initial value problem and the periodic boundary value problem, the unique continuation results found in \cite{LiPo} are optimal. More precisely, the result established there for the constant $c_0=0$ fails for any constant $c_0\neq 0$.