Interactions and Focusing of Nonlinear Water Waves

A theoretical and computational study is undertaken for the modulational instabilities of a pair of nonlinearly interacting two-dimensional waves in deep water. It has been shown that the full dynamics of these interacting waves gives rise to localized large-amplitude wavepackets (wave focusing). The coupled cubic nonlinear Schr¨odinger (CNLS) equations are used to derive a nonlinear dispersion equation which give rise to new class of modulational instabilities and demonstrates the dependence of obliqueness of the interacting waves. The computations, due to nonlinear wave-wave interactions, waves that are separately modulationally stable can give rise to the formation of largeamplitude coherent wave packets with amplitudes several times that of the initial waves. In the case for the original Benjamin-Feir instability, in contrast, waves disintegrate into a wide spectrum.