An interesting phenomenon in water channels is the appearance of waves with length much greater than the depth of the water. Recently, R. Camassa and D. Holm proposed a new model for the same phenomenon:
The variable in (1) represents the fluid velocity at time in the direction in appropriate nondimensional units (or, equivalently, the height of the water’s free surface above a flat bottom).
The aim of this paper is to prove local well-posedness of strong solutions to (1) for a large class of initial data, and to analyze global existence and blow-up phenomena. In addition, we introduce the notion of weak solutions to (1) suitable for soliton interaction.