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Emergent rogue wave structures and statistics in spontaneous modulation instability

Abstract : The nonlinear Schrödinger equation (NLSE) is a seminal equation of nonlinear physics describing wave packet evolution in weakly-nonlinear dispersive media. The NLSE is especially important in understanding how high amplitude " rogue waves " emerge from noise through the process of modulation instability (MI) whereby a perturbation on an initial plane wave can evolve into strongly-localised " breather " or " soliton on finite background (SFB) " structures. Although there has been much study of such structures excited under controlled conditions, there remains the open question of how closely the analytic solutions of the NLSE actually model localised structures emerging in noise-seeded MI. We address this question here using numerical simulations to compare the properties of a large ensemble of emergent peaks in noise-seeded MI with the known analytic solutions of the NLSE. Our results show that both elementary breather and higher-order SFB structures are observed in chaotic MI, with the characteristics of the noise-induced peaks clustering closely around analytic NLSE predictions. A significant conclusion of our work is to suggest that the widely-held view that the Peregrine soliton forms a rogue wave prototype must be revisited. Rather, we confirm earlier suggestions that NLSE rogue waves are most appropriately identified as collisions between elementary SFB solutions. The terminology of " rogue wave " in physics describes events with high amplitude that emerge randomly in the dynamical behaviour of a particular system with low probability. This label was initially applied to describe the unexpected appearance of large and destructive waves on the ocean 1,2 but has now been generalized to describe large amplitude rare events in many other systems 3,4. Particular interest in rogue waves emerging during propagation in systems described by the nonlinear Schrödinger equation (NLSE) or its extensions has led to studies of rogue wave behaviour for deep water wave groups, pulse propagation in optical fibres, plasmas and cold atoms 5–7. The NLSE has particular significance in the context of rogue wave behaviour because it exhibits the Benjamin-Feir or modulation instability (MI), where a weak modulation on a plane wave will undergo exponential growth with propagation 8,9. After this initial exponential growth, the subsequent dynamics sees periodic growth and decay in a form of Fermi-Pasta-Ulam (FPU) recurrence 10. Because rapid growth and decay of a weakly modulated pulse envelope would also increase the amplitude and steep-ness of an underlying carrier wave, MI has long been considered a primary candidate for a rogue wave generating mechanism 11–13. Although the initial mathematical studies of MI were performed using linear stability analysis 9 , MI and FPU dynamics in the NLSE can also be described using various types of " breather " or soliton on finite background (SFB) solutions to the NLSE 14–17. The possibility to describe these dynamics analytically has motivated much research to obtain possible insights into the particular initial conditions that
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Shanti Toenger, Thomas Godin, Cyril Billet, Frédéric Dias, Miro Erkintalo, et al.. Emergent rogue wave structures and statistics in spontaneous modulation instability. Scientific Reports, Nature Publishing Group, 2015, 5 (1), ⟨10.1038/srep10380⟩. ⟨hal-01795553⟩

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