On nonlinear evolution equations of a three-dimensional capillary-gravity wave packet at the interface of two superposed fluids

Abstract

Assuming amplitudes as slowly varying functions of space and time and using a perturbation method, two coupled nonlinear partial differential equations are derived that give the nonlinear evolution of the amplitude of a three-dimensional capillary-gravity wave packet at the interface of two superposed incompressible fluid layers of finite depths, including the effect of its interaction with a long gravity wave. Starting from these two coupled equations, a balanced set of modulation equations, both at nonresonance and at resonance, is derived. The balanced set of modulation equations, at nonresonance, reduces to a single nonlinear Schrödinger equation, if it is assumed that space variation of the amplitudes depends only on variation along an arbitrary fixed horizontal direction. Modulational instability conditions, both at resonance and at nonresonance, are also deduced. The advantage of the perturbation method adopted in the present problem, over the reductive perturbation method, is noticed.