PERIODIC-SOLUTIONS FOR A COMPLEX HAMILTONIAN SYSTEM - NEW STANDING WATER-WAVES

The interaction of several (up to 39) two-dimensional, deep water, gra vity, standing wave modes is studied to high order (up to 18) using a Hamiltonian formulation. Four new time-periodic solutions are found. T he stability of periodic standing waves to superharmonic perturbations is studied. A new instability of the standing Stokes wave is found at moderate steepness (0.266). Two of the new waves are weakly unstable. Stability of the Stokes wave to subharmonic perturbation is also stud ied. The long time evolution of the perturbed waves is examined. The H amiltonian method gives most accurate results. Recurrence of the unsta ble Stoke waves is revealed. The periodic solutions are useful for und erstanding the dynamical system of the standing wave modes.