High-order formulation of the water-wave problem

The water-wave problem has a Hamiltonian formulation as derived by Zakharov. This leads to coupled nonlinear evolution equations for discrete wave modes. Discrete spectra occur naturally for standing waves in a basin. For high enough order, high enough energy and a sufficient number of degrees of freedom, chaos may emerge. We have derived the Hamiltonian and evolution equations to arbitrary order. The conditions for the convergence of the expansion of the Hamiltonian are discussed. We performed numerical studies of the interaction of two standing wave modes for various cases and for increasing order. Using Poincaré sections, the onset of local chaos in resonance and near-resonance conditions is manifested. Chaos appears at low order models but vanishes with increasing order. The significance of the newly derived high-order formulation for water waves is clearly demonstrated.