Geometrical demonstration for persistence properties for a bi-Hamiltonian shallow water system

TL;DR

This paper provides a geometric proof of persistence properties for a bi-Hamiltonian shallow water system, enhancing understanding of wave behavior and unique continuation in both periodic and non-periodic cases.

Contribution

It introduces a novel geometric approach using Hamiltonians to demonstrate persistence and unique continuation properties in shallow water models.

Findings

Improved results on unique continuation properties.

Demonstrated that the only compactly supported solution is zero.

Extended analysis to both periodic and non-periodic cases.

Abstract

We present a geometrical demonstration for persistence properties for a bi-Hamiltonian system modelling waves in a shallow water regime. Both periodic and non-periodic cases are considered and a key ingredient in our approach is one of the Hamiltonians of the system. As a consequence of our developments we improve recent works dealing with unique continuation properties for shallow water equations, as well as we provide a novel way to prove that the unique compactly supported solution of the system is necessarily the zero function.