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

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.