Conserved quantities, continuation and compactly supported solutions of some shallow water models

TL;DR

This paper proves a unique continuation property for solutions of certain shallow water models, showing that solutions vanishing on an open set must be identically zero, using conserved quantities and a geometric approach.

Contribution

It introduces a novel geometric method leveraging conserved quantities to establish unique continuation for Camassa-Holm type equations.

Findings

Solutions vanishing on an open set are identically zero

The method applies to a broad class of equations of the Camassa-Holm type

Provides a new perspective on the structure of solutions for shallow water models

Abstract

A proof that strong solutions of the Dullin-Gottwald-Holm equation vanishing on an open set of the (1+1) space-time are identically zero is presented. In order to do it, we use a geometrical approach based on the conserved quantities of the equation to prove a unique continuation result for its solutions. We show that this idea can be applied to a large class of equations of the Camassa-Holm type.