UNCORRECTED PROOF Physica D 3011 (2002) 1–20 Nonlinear wave focusing on water of finite depth 3 A. Slunyaev a , C. Kharif b,∗ , E. Pelinovsky a , T. Talipova a 4 a Laboratory of Hydrophysics and Nonlinear Acoustics, Institute of Applied Physics, Nizhny Novgorod, Russia 5 b Institut de Recherche sur les Phénomènes Hors Equilibre, 49 rue F. Joliot-Curie, B.P. 146, 13384 Marseille Cedex 13, France 6 Received 29 March 2002; received in revised form 5 August 2002; accepted 2 September 2002 7 Communicated by S. Fauve 8 Abstract 9 The problem of freak wave formation on water of finite depth is discussed. Dispersive focusing in a nonlinear medium is 10 suggested as a possible mechanism of giant wave generation. This effect is considered within the framework of the nonlinear 11 Schrödinger equation and the Davey–Stewartson system, describing 2 + 1-dimensional surface wave groups on water of finite 12 depth. In the 2 + 1-dimensional case, the dispersive grouping is accompanied with a geometrical focusing. Necessary wave 13 conditions for the occurrence of such a phenomenon are discussed. Influence of non-optimal phase modulation and presence 14 of strong random wave component are found to be weak: they do not cancel the mechanism of wave amplification. The 15 mechanism of dispersive focusing is compared with the wave enhancement due to the Benjamin–Feir instability, which is 16 found to be extremely sensitive with respect to weak random perturbations. 17 © 2002 Published by Elsevier Science B.V. 18 PACS: 92.10.Hm; 47.35.+i; 92.60.Dj 19 Keywords: Freak waves; Rogue waves; Dispersive focusing; Nonlinear focusing; Modulational instability; Davey–Stewartson equations 20 1. Introduction 21 The freak phenomenon in the ocean has become a topic of intensive study during recent years. This event is the 22 rapid occurrence of an unexpected huge wave. Such a wave may lead to damage of ships and offshore platforms and 23 to deaths. Many registered cases can be found in [1–3]. Several theories were suggested as possible explanations for 24 this phenomenon. Some of them use a strong current [3–5], which may amplify the wave. But since the phenomenon 25 was registered in different areas of the World Ocean, without strong currents, other possible theories were suggested. 26 Response to the question: What is the expected surface configuration surrounding an extreme wave crest, can be 27 found in [6,7]. The theoretical predictions based on [8] were confirmed by field experiments [9] and buoy data 28 from SWADE. Dispersive wave focusing is well known in linear theory and this mechanism was considered for 29 the understanding of the freak wave phenomenon some time ago [10–13]. But the question of how nonlinearity of 30 the medium may affect this mechanism has not been definitely answered. It is obvious, that linear wave focusing 31 may be destroyed by detuning due to nonlinear interaction of the Fourier harmonics of the wave field. This problem 32 ∗ Corresponding author. E-mail address: kharif@irphe.univ-mrs.fr (C. Kharif). 1 0167-2789/02/$ – see front matter © 2002 Published by Elsevier Science B.V. 2 PII:S0167-2789(02)00662-0