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Extreme wave events, also referred to as freak or rogue waves, are mostly known as oceanic phenomena responsible for a large number of maritime disasters. These waves, which have height and steepness much greater than expected from the sea average state, have recently become a topic of intense research. Freak waves appear both in deep ocean and in shallow water. The danger of oceanic rogue waves is that they suddenly appear from nowhere only seconds before they hit a ship. The grim reality, however, is that although the existence of freak waves has now been confirmed by multiple observations, uncertainty remains on their fundamental origins.
A formal mathematical description of a rogue wave is provided by the so-called Peregrine soliton. This solitary wave is a solution of the scalar nonlinear Schroedinger equation (NLSE) with the property of being localized in both coordinates: thus it describes a unique wave event (see figure on the left). In a variety of complex systems such as Bose-Einstein condensates, optical fibers, and financial systems, several amplitudes rather than a single one need to be considered. The resulting systems of coupled equations may thus describe extreme waves with higher accuracy than the scalar NLSE model. We constructed new multiparametric vector soliton solutions of the vector NLSE equations (figure below). A key novel property of this solution is that it features both exponential and rational dependence on coordinates and, therefore, it is called semirational. Such extreme wave may be observed for example in fiber optic communication systems using the polarization multiplexing technique for doubling the transmission capacity.
Shallow water rogue waves in optical fibers with normal dispersion
Traditionally, rogue waves are linked with the presence of modulation instability (MI), whose nonlinear development is described by the so-called Akhmediev breathers. Rogue waves in optical fibers may also be generated in the normal group-velocity dispersion (GVD) regime of pulse propagation, where MI is absent. Indeed, nonlinearity driven pulse shaping in this case may be described in terms of the semi-classical approximation to the NLSE, which leads to the so-called nonlinear shallow water equation (NSWE). Therefore we have established a direct link between the dynamics of extreme wave generation in shallow waters, such as tsunamis, and their direct counterparts in optical communication systems. Since the CW state of the field is stable, shallow water optical rogue waves may only be generated as a result of particular setting of the initial or boundary conditions. Namely, the initial modulation of the optical frequency, which is analogous to considering the collision between oppositely directed currents near the beach, or the merging of different avalanches falling from a mountain valley. The figures below illustrate the dynamics of the generation of an intense, flat-top, self-similar and chirp-free pulse as a result of the initial step-wise frequency modulation of a CW laser. In hydrodynamics, this corresponds to the hump of water which is generated by two water waves traveling with opposite velocities. The intriguing property of such pulses, that we call flaticons, is their stable merging upon mutual collision into either a steady or transient high-intensity wave. The pulse collision dynamics may also lead to the formation of extreme intensity peaks in optical communication systems whenever various wavelength channels are transported on the same fiber.
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