Effect of local Peregrine soliton emergence on statistics of random waves in the one-dimensional focusing nonlinear Schr\"odinger equation

Effect of local Peregrine soliton emergence on statistics of random waves in the one-dimensional focusing nonlinear Schrödinger equation

Alexey Tikan

Phys. Rev. E 101, 012209 – Published 14 January 2020

Abstract

The Peregrine soliton is often considered as a prototype of rogue waves. After recent advances in the semiclassical limit of the one-dimensional focusing nonlinear Schrödinger equation [M. Bertola and A. Tovbis, Commun. Pure Appl. Math. 66, 678 (2013)] this conjecture can be seen from another perspective. In the present paper, connecting deterministic and statistical approaches, we numerically demonstrate the effect of the universal local emergence of Peregrine solitons on the evolution of statistical properties of random waves. Evidence of this effect is found in recent experimental studies in the contexts of fiber optics and hydrodynamics. The present approach can serve as a powerful tool for the description of the transient dynamics of random waves and provide new insights into the problem of the rogue waves formation.

Received 4 June 2019
Revised 27 October 2019

DOI:https://doi.org/10.1103/PhysRevE.101.012209

Physics Subject Headings (PhySH)

Nonlinear optics

Breathers Nonlinear waves

Nonlinear DynamicsStatistical Physics & ThermodynamicsFluid DynamicsAtomic, Molecular & Optical

Authors & Affiliations

Alexey Tikan ^*

University of Lille, UMR 8523-PhLAM-Physique des Lasers Atomes et Molecules, F-59000 Lille, France

^*alexey.tikan@univ-lille.fr

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Vol. 101, Iss. 1 — January 2020

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Figure 1
Evolution of the PS. (a) Spatiotemporal diagram of $| ψ |$ plotted according the analytical formula [9]. (b) Cross section at $ξ = 0$ . (c) Corresponding phase.
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Figure 2
Regularization of the gradient catastrophe by local emergence of the PS. Parameter $ε$ in the simulation is 1/10. (a)–(c) Spatiotemporal diagram, amplitude, and phase cross section, respectively, at the maximum compression point. The $10 sech (τ)$ function is taken as an initial condition. The black dashed line represents a fit of the local PS with the analytical formula (6). The maximum compression point in the simulation occurs at the distance $ξ = 0.673$ , while the distance predicted by expression (5) is 0.6514.
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Figure 3
Numerical simulations of a partially coherent wave propagation in the focusing NLS system. (a) Probability density function of ${| ψ |}^{2}$ at three different propagation distances: $ξ = 0$ (green), 0.366 (orange), and 2 (red). The dashed black line corresponds to $exp (- | ψ |^{2})$ . (b) Spatiotemporal diagram of $| ψ |$ . White boxes highlight the coherent structures that appear out of initial humps and can be locally fitted with the PS. (c) Evolution of the kurtosis (black). Green, orange, and red lines correspond to $ξ = 0$ , 0.366, and 2. The kurtosis and the probability density function are computed over 10 000 realizations of the partially coherent wave similar to one depicted in panel (b). (d) ${| ψ |}^{2}$ profile at two values of $ξ = 0$ , 0.366 (colors are preserved). Inset plot demonstrates a fit of the high-amplitude event with the exact formula of the PS. In all the simulations the value of $ε$ is 0.2.
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Figure 4
Prediction of the maximum of the kurtosis using the Bertola-Tovbis approach. Comparison between the probability density of local PS soliton emergence point (red dot, right axis) and the kurtosis at different propagation distances (blue line, left axis). All parameters coincide with ones in Fig. 3.
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Figure 5
Numerical simulations with real optical parameters. (a) and (b) Comparison between the probability density of local PS emergence point (red dot, right axis) and the kurtosis at different propagation distances (blue line, left axis). Black dashed line corresponds to the kurtosis of the partially coherent signal with zero phase. Spectral width of the partially coherent initial conditions 0.1 (a) and 0.2 (b) $THz$ , average power is 2.6 $W$ for both cases, $β_{2} = - 22 {ps}^{2} / km$ , and $γ = 2.4 {(W km)}^{- 1}$ . These parameters correspond to $ε$ 0.26 and 0.53, respectively. (c) Prediction for the kurtosis maximum position for different values of spectral width keeping all other parameters fixed. Covered range of $ε$ varies from 0.053 to 0.53.
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Physical Review E

covering statistical, nonlinear, biological, and soft matter physics

Effect of local Peregrine soliton emergence on statistics of random waves in the one-dimensional focusing nonlinear Schrödinger equation

Alexey Tikan

Phys. Rev. E 101, 012209 – Published 14 January 2020

Abstract

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Physical Review E

covering statistical, nonlinear, biological, and soft matter physics

Effect of local Peregrine soliton emergence on statistics of random waves in the one-dimensional focusing nonlinear Schrödinger equation

Alexey Tikan

Phys. Rev. E 101, 012209 – Published 14 January 2020

Abstract

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