Strongly interacting soliton gas and formation of rogue waves

Abstract

We study numerically the properties of (statistically) homogeneous soliton gas depending on soliton density (proportional to number of solitons per unit length) and soliton velocities, in the framework of the focusing one-dimensional Nonlinear Schrödinger (NLS) equation. In order to model such gas we use N -soliton solutions (N -SS) with N ∼ 100, which we generate with specific implementation of the dressing method combined with 100-digits arithmetics. We examine the major statistical characteristics, in particular the kinetic and potential energies, the kurtosis, the wave-action spectrum and the probability density function (PDF) of wave intensity.We show that in the case of small soliton density the kinetic and potential energies, as well as the kurtosis, are very well described by the analytical relations derived without taking into account soliton interactions. With increasing soliton density and velocities, soliton interactions enhance, and we observe increasing deviations from these relations leading to increased absolute values for all of these three characteristics. The wave-action spectrum is smooth, decays close to exponentially at large wavenumbers and widens with increasing soliton density and velocities. The PDF of wave intensity deviates from the exponential (Rayleigh) PDF drastically for rarefied soliton gas, transforming much closer to it at densities corresponding to essential interaction between the solitons. Rogue waves emerging in soliton gas are multi-soliton collisions, and yet some of them have spatial profiles very similar to those of the Peregrine solutions of different orders. We present example of three-soliton collision, for which even the temporal behavior of the maximal amplitude is very well approximated by the Peregrine solution of the second order.

Rogue waves on the double-periodic background in the focusing nonlinear Schrödinger equation

“…Their appearance is related to the modulation instability of the wave background [35]. Formation of particular rogue waves such as Akhmediev breathers, Peregrine solution, and Kuznetsov-Ma breathers have been modeled from different initial data such as local condensates [22], multi-soliton gases [23,34], and periodic perturbations [2,3]. Experimental observations of rogue waves have been confirmed both in hydrodynamical and optical laboratories [28,5].…”

“…Experimental observations of rogue waves have been confirmed both in hydrodynamical and optical laboratories [28,5]. Statistical analysis of rogue waves has recently been developed [29] (see also [2,23]).…”

Nonlinear Spectral Synthesis of Soliton Gas in Deep-Water Surface Gravity Waves

“…We take advantage of the recently developed methodol-ogy for the effective numerical construction of the socalled N -soliton solutions of the focusing 1D-NLSE with N large (ref. [48]), to create an incoherent wavefield having a dominant and controlled solitonic content characterized by a measurable DOS. We show that the generated SG may undergo some complex space-time evolution while the discrete IST spectrum is found to be nearly conserved, albeit being perturbed by higher-order effects.…”

“…As shown in ref. [48,51], such N -soliton statistical ensemble is a good model for a homogeneous dense SG.…”

Breather’s Properties within the Framework of the Modified Korteweg–de Vries Equation

“…Then, a broad study of soliton turbulence using numerical methods in optics and hydrodynamics began [4][5][6][7][8][9]. Obviously, wave structures with very high energy, called freak waves, can appear resulting from many particle (soliton) collisions [10][11][12][13]. Many books and reviews are devoted to this phenomenon in nonlinear physics [14,15].…”