ABSTRACT
Nonlinear wave focusing originating from the universal modulation instability (MI) is responsible for the formation of strong wave localizations on the water surface and in nonlinear wave guides, such as optical Kerr media and plasma. Such extreme wave dynamics can be described by breather solutions of the nonlinear Schrödinger equation (NLSE) like by way of example the famed doubly-localized Peregrine breathers (PB), which typify particular cases of MI. On the other hand, it has been suggested that the MI relevance weakens when the wave field becomes broadband or directional. Here, we provide experimental evidence of nonlinear and distinct PB-type focusing in standing water waves describing the scenario of two counterpropagating wave trains. The collected collinear wave measurements are in excellent agreement with the hydrodynamic coupled NLSE (CNLSE) and suggest that MI can undisturbedly prevail during the interplay of several wave systems and emphasize the potential role of exact NLSE solutions in extreme wave formation beyond the formal narrow band and unidirectional limits. Our work may inspire further experimental investigations in various nonlinear wave guides governed by CNLSE frameworks as well as theoretical progress to predict strong wave coherence in directional fields.
ABSTRACT
We study the extended Korteweg-de Vries equation, that is, the usual Korteweg-de Vries equation but with the inclusion of an extra cubic nonlinear term, for the case when the coefficient of the cubic nonlinear term has an opposite polarity to that of the coefficient of the linear dispersive term. As this equation is integrable, the number and type of solitons formed can be determined from an appropriate spectral problem. For initial disturbances of small amplitude, the number and type of solitons generated is similar to the well-known situation for the Korteweg-de Vries equation. However, our interest here is in initial disturbances of larger amplitude, for which there is the possibility of the generation of large-amplitude "table-top" solitons as well as small-amplitude solitons similar to the solitons of the Korteweg-de Vries equation. For this case, and in contrast to some earlier results which assumed that an initial disturbance in the shape of a rectangular box would be typical, we show that the number and type of solitons formed depend crucially on the disturbance shape, and change drastically when the initial disturbance is changed from a rectangular box to a "sech"-profile. (c) 2002 American Institute of Physics.
