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Register a new userObservation of modulation instability and rogue breathers on stationary periodic waves
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We present both theoretical description and experimental observation of the modulation instability process and related rogue breathers in the case of stationary periodic background waves, namely cnoidal and dnoidal envelopes. Despite being well-known solutions of the nonlinear Schrodinger equation, the stability of such background waves has remained unexplored experimentally until now, unlike the fundamental plane wave. By means of two experimental setups, namely, in nonlinear optics and hydrodynamics, we report on quantitative measurements of spontaneous modulation instability gain seeded by input random noise, as well as the formation of rogue breather solutions induced by a coherent perturbation. Our results confirm the generalization of modulation instability when more complex background waves are involved.
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Rogue waves are abnormally large waves which appear unexpectedly and have attracted considerable attention, particularly in recent years. The one space, one time (1+1) nonlinear Schrodinger equation is often used to model rogue waves; it is an envelope description of plane waves and admits the so-called Pergerine and Kuznetov-Ma soliton solutions. However, in deep water waves and certain electromagnetic systems where there are two significant transverse dimensions, the 2+1 hyperbolic nonlinear Schrodinger equation is the appropriate wave envelope description. Here we show that these rogue wave solutions suffer from strong transverse instability at long and short frequencies. Moreover, the stability of the Peregrine soliton is found to coincide with that of the background plane wave. These results indicate that, when applicable, transverse dimensions must be taken into account when investigating rogue wave pheneomena.
We report an exact link between Zakharov-Gelash super-regular (SR) breathers (formed by a pair of quasi-Akhmediev breathers) with interesting different nonlinear propagation characteristics and modulation instability (MI). This shows that the absolute difference of group velocities of SR breathers coincides exactly with the linear MI growth rate. This link holds for a series of nonlinear Schr{o}dinger equations with infinite-order terms. For the particular case of SR breathers with opposite group velocities, the growth rate of SR breathers is consistent with that of each quasi-Akhmediev breather along the propagation direction. Numerical simulations reveal the robustness of different SR breathers generated from various non-ideal single and multiple initial excitations. Our results provide insight into the MI nature described by SR breathers and could be helpful for controllable SR breather excitations in related nonlinear systems.
Experimental results describing random, uni-directional, long crested, water waves over non-uniform bathymetry confirm the formation of stable coherent wave packages traveling with almost uniform group velocity. The waves are generated with JONSWAP spectrum for various steepness, height and constant period. A set of statistical procedures were applied to the experimental data, including the space and time variation of kurtosis, skewness, BFI, Fourier and moving Fourier spectra, and probability distribution of wave heights. Stable wave packages formed out of the random field and traveling over shoals, valleys and slopes were compared with exact solutions of the NLS equation resulting in good matches and demonstrating that these packages are very similar to deep water breathers solutions, surviving over the non-uniform bathymetry. We also present events of formation of rogue waves over those regions where the BFI, kurtosis and skewness coefficients have maximal values.
The nonlinear stage of modulational instability in optical fibers induced by a wide and easily accessible class of localized perturbations is studied using the nonlinear Schrodinger equation. It is showed that the development of associated spatio-temporal patterns is strongly affected by the shape and the parameters of the perturbation. Different scenarios are presented that involve an auto-modulation developing in a characteristic wedge, possibly coexisting with breathers which lie inside or outside the wedge.
In the present work, we aim at taking a step towards the spectral stability analysis of Peregrine solitons, i.e., wave structures that are used to emulate extreme wave events. Given the space-time localized nature of Peregrine solitons, this is a priori a non-trivial task. Our main tool in this effort will be the study of the spectral stability of the periodic generalization of the Peregrine soliton in the evolution variable, namely the Kuznetsov--Ma breather. Given the periodic structure of the latter, we compute the corresponding Floquet multipliers, and examine them in the limit where the period of the orbit tends to infinity. This way, we extrapolate towards the stability of the limiting structure, namely the Peregrine soliton. We find that multiple unstable modes of the background are enhanced, yet no additional unstable eigenmodes arise as the Peregrine limit is approached. We explore the instability evolution also in direct numerical simulations.
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