Intermittency in generalized NLS equation with focusing six-wave interactions

Abstract

We study numerically the statistics of waves for generalized one-dimensional Nonlinear Schrödinger (NLS) equation that takes into account focusing six-wave interactions, dumping and pumping terms. We demonstrate the universal behavior of this system for the region of parameters when six-wave interactions term affects significantly only the largest waves. In particular, in the statistically steady state of this system the probability density function (PDF) of wave amplitudes turns out to be strongly non-Rayleigh one for large waves, with characteristic "fat tail" decaying with amplitude |Ψ| close to ∝ exp(−γ|Ψ|), where γ > 0 is constant. The corresponding non-Rayleigh addition to the PDF indicates strong intermittency, vanishes in the absence of six-wave interactions, and increases with six-wave coupling coefficient.

Integrable turbulence generated from modulational instability of cnoidal waves

“…14had duration in time ∆T ∼ 0.5 and reached maximal amplitude max |Ψ| = 4.4 at t = 1361.4, that is sufficiently close to the asymptotic stationary state. The crest of this wave corresponds to relative intensity I = 29.We examined several hundreds of rogue waves detected in our experiments, and found that all of them at the time of their maximal elevation have quasi-rational profile similar to that of the Peregrine solution[31,32] (see the similar results in[10] for incoherent wave initial conditions and in[9] for generalized NLS equation accounting for six-wave interactions, pumping and dumping terms). The Peregrine solution is localized in space and time algebraic solution of the (Color on-line) (a) Space distribution of amplitude |Ψ(x, t 0 )| (solid black) and phase arg Ψ(x, t 0 ) (dash-dot red) for the largest detected rogue wave at the time t 0 = 1361.4 of its maximal elevation.…”

“…However, for cn-branch some properties of the integrable turbulence should be different. For instance, we expect that cnoidal waves (5) with large ω 1 will lead to almost linear integrable turbulence with very small asymptotic potential to kinetic energy ratio |Q A | 1, since these cnoidal waves are close to sinusoidal wave with exponentially small amplitude (9). Therefore, for cn-branch the asymptotic energy ratio Q A should not be fixed to -2, but should instead vary from Q A → −2 for ω 1 → 0 to Q A → 0 for ω 1 → +∞.…”

“…The statistics of waves for different nonlinear systems was intensively studied in the recent years [1][2][3][4][5][6][7][8][9][10], especially since the first experimental observation of optical rogue waves [11].…”

Multisoliton interactions approximating the dynamics of breather solutions

“…Also, they describe rogue waves that appear either once or periodically in time. However, in numerical simulations, breather‐like rogue waves appear randomly from time to time, and this happens in various scenarios, for instance, long after the MI is fully developed, 34,68,83 or in the long‐time evolution from partially coherent waves, 35 or in nearly integrable systems 84 . The existing breather models are incapable to capture this behavior, but it can be adequately described by soliton gas models.…”

Comparison of Split-Step and Hamiltonian Integration Methods for Simulation of the Nonlinear Schrödinger Equation

“…The accurate portrayal of soliton interactions is crucial for the simulation of interactions in soliton gas [27,28,29,30], or the fast-developing field of integrable turbulence [3], both methods are well-suited for this. At the same time one should mention that split-step is simpler to implement and is more efficient memory-wise.…”