Aspects of a shallow water equation

We discuss a recently derived model for the propagation of waves in shallow water. The equation is integrable and its solitary waves are solitons. The only way a singularity can arise in a classical solution is in the form of wave-breaking. The nonlinear PDE can be viewed as geodesic flow on the manifold of diffeomorphisms of the line. This geometric interpretation is very useful in dealing with qualitative aspects of the equation.