Distributional Profiles for Traveling Waves in the Camassa-Holm Equation

In this paper travelling waves with distributional profiles for the Camassa-Holm equation are studied. Using a product of distributions a new solution concept is introduced which extends the classical one. As a consequence, necessary and sufficient conditions for the propagation of a distributional profile are established and we present examples with discontinuous solutions, measures and even distributions that are not measures. One of these examples may be interpreted as a simple model for a "tsunami" in the setting of shallow water theory. We also prove that, under natural assumptions, profiles belonging to the Sobolev space H-loc(1)(R) usually considered in the classical weak formulation can be seen as particular cases of our distributional solution concept.