Nonlinear Parabolic Equation and Extreme Waves on the Sea Surface

Abstract

Nearly 40 years have passed since V. I. Talanov discovered the nonlinear parabolic equation which played an important role in the nonlinear optics. It was very quickly understood that this equation could also be adapted for nonstationary wave packets of different physical nature and of any dimension. Under the later name of the nonlinear (cubic) Schrodinger equation, it became a fundamental equation in the theory of weakly nonlinear wave packets in media with strong dispersion. The article is devoted to only one application of the nonlinear Schrodinger equation in the theory of the so-called freak waves on the sea surface. In the last five years a great boom has occurred in the research of extreme waves on the water, for which the nonlinear parabolic equation played an important role in the understanding of physical mechanisms of the freak-wave phenomenon. More accurate, preferably numerical, models of waves on a water with more comprehensive account of the nonlinearity and dispersion come on the spot today, and many results of weakly nonlinear models are already corrected quantitatively. Nevertheless, sophisticated models do not bring new physical concepts. Hence, their description on the basis of the nonlinear parabolic equation (nonlinear Schrodinger equation), performed in this paper, seems very attractive in view of their possible applications in the wave-motion physics.