Asymptotic Properties Of The Solutions Of The Camassa-Holm Equation

This artical is about the asymptotic properties of Camassa-Holm (CH) equation. This paper proves that the asymptotic density of the positive momentum density of the CH equation is a combination of Dirac measures supported on the positive axis. This means that as time goes to infinity, momentum density concentrates in small intervals moving right with different constant speeds. The solution essentially decays exponentially outside these intervals.From the existing literature as well as numerical results and special cases of explicit solutions, there are evidences that many initial conditions of CH equation give rise to a train of solitary waves moving at different speeds. In this thesis, using the method of asymptotic density, under the assumption that it is unique, we give further evidences for this phenomenon. Similar discussions are valid for the Degasperis-Procesi equation.