Modulational instabilities and Fermi-Pasta-Ulam recurrence in a coupled long wave-short wave system, with a mismatch in group velocity

The resonance of two envelopes of short (capillary) waves with a common long (gravity) wave component is considered. A mismatch in group velocity is incorporated and the resonance conditions need not be satisfied exactly. This slight detuning permits a wider choice of modes and consequently, a much richer set of dynamics. The linear instability of plane waves is studied, and the dominant unstable wave numbers are identified. The subsequent fully nonlinear evolution of these perturbed plane waves is investigated by direct numerical simulations. The Fermi – Pasta – Ulam recurrence phenomenon is observed.