Nonlinear wave focusing on water of finite depth. (English) Zbl 1047.76010
Summary: The problem of freak wave formation on water of finite depth is discussed. Dispersive focusing in a nonlinear medium is suggested as a possible mechanism of giant wave generation. This effect is considered within the framework of nonlinear Schrödinger equation and Davey-Stewartson system describing \(2+1\)-dimensional surface wave groups on water of finite depth. In the \(2+1\)-dimensional case, the dispersive grouping is accompanied with a geometrical focusing. Necessary wave conditions for the occurrence of such a phenomenon are discussed. Influence of non-optimal phase modulation and presence of strong random wave component are found to be weak: they do not cancel the mechanism of wave amplification. The mechanism of dispersive focusing is compared with the wave enhancement due to Benjamin-Feir instability, which is found to be extremely sensitive with respect to weak random perturbations.
MSC:
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 76E99 | Hydrodynamic stability |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
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