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 $\mu$ is the small parameter corresponding to the inverse of the wave length, we show that the normal form at order ${\mu }^5$ consists of two decoupled equation, one describing right going waves and the other describing left going waves. Performing a further non Hamiltonian transformation we conjugate each of these equations to a linear combination of the first three equations in the KdV hierarchy. At order ${\mu }^7$ we find nontrivial terms coupling the two counter-propagating waves.