The classical shallow water equations: Symplectic geometry

The classical shallow water equations express the change with time of the height h and the velocity ν of a 1-dimensional fluid: νξ νt+ νξ νx+ νh νx=0. νh νx+ νhν νx=0. They possess an infinite number of integrals of motion due to Benney [1973] and can be written in Hamiltonian form relative to a symplectic structure introduced by Manin [1978]. The present paper deals with their complete integrability up to the advent of shocks. This is proved in the small under an extra assumption satisfied by most height-velocity pairs: that hh′ = ± ν′ only at isolated points.