Hamiltonian Studies on Counter-Propagating Water Waves

We use a Hamiltonian normal form approach to study the dynamics of the water wave problem in the small-amplitude long-wave regime (KdV regime). If is the small parameter corresponding to the inverse of the wave length, we show that the normal form at order 5 consists of two decoupled equations: one describing right going waves and the other describing left going waves. Each of these equations is integrable: it is a linear combination of the first three equations in the KdV hierarchy. At order 7, we find nontrivial terms coupling the two counter-propagating waves.