Weakly two-dimensional interaction of solitons in shallow water

Abstract

Nonlinear interactions of long-crested solitonic waves travelling in different directions in shallow water may serve as a possible mechanism of freak waves in certain sea areas. Several features of such interactions of equal amplitude Korteweg–de Vries (KdV) solitons in the framework of the Kadomtsev–Petviashvili equation are reviewed. In certain cases, nonlinear coupling produces a particularly high and steep wave hump. Interactions of equal amplitude solitons may lead to water surface elevations up to four times as high as the amplitude of the counterparts and the slope of the wave front may encounter eightfold increase. Exact expressions for the maximum slope in the case of interactions of unequal amplitude solitons are derived. The slope amplification for a certain class of interactions in the limiting case is twice as intense as the amplitude amplification. In the limiting case of exact resonance the interaction pattern is a new KdV soliton as in the case of the Mach stem. This feature allows to directly establish the extreme properties of the humps excited by nonlinear coupling. Evidence of such interactions and their possible consequences in realistic conditions are discussed.