Fermi-Pasta-Ulam-Tsingou recurrence of periodic anomalous waves in the complex Ginzburg-Landau and in the Lugiato-Lefever equations

Coppini, Fabio; Santini, P. M.

doi:10.1103/physreve.102.062207

Cited by 15 publications

(18 citation statements)

References 52 publications

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“…The AB is unstable with respect to perturbations of the focusing NLS equation, as it was shown in real and numerical experiments [42,67], and as it was analytically proven in [24], where the NLS perturbation theory for AWs was constructed and applied to the case of the NLS perturbed by linear loss or gain terms, and in [25], where such a theory was applied to the complex Ginzburg-Landau [58] and Lugiato Lefever [51] equations, viewed as perturbations of focusing NLS.…”

Section: Conclusion Open Problems and Remarksmentioning

confidence: 84%

“…It is well established that they are unstable with respect to small perturbations of the NLS equation [42,67,24,25]; see also a finite-gap model describing the numerical instabilities of the AB [37].…”

Section: Introductionmentioning

confidence: 99%

“…The solution [34,35] of the NLS periodic Cauchy problem (4) of the AWs is based on the proper adaptation of the finite-gap method [59,28,46,49,55,43] (see [45] for its first application to NLS), to it. See also [36] for an alternative approach to the study of the AW recurrence, based on matched asymptotic expansions; see [38] for the analytic study of the phase resonances in the AW recurrence; see [65], [26] and [27] for the analytic study of the AW recurrence in other NLS type models: respectively the PT-symmetric NLS equation [5], the Ablowitz-Ladik model [4], and the massive Thirring model [69,56].…”

Section: Introductionmentioning

confidence: 99%

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The linear and nonlinear instability of the Akhmediev breather

Grinevich,

Santini

2020

Preprint

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The Akhmediev breather (AB) and its M-soliton generalization, hereafter called AB M , are exact solutions of the focusing NLS equation periodic in space and exponentially localized in time over the constant unstable background; they describe the appearance of M unstable nonlinear modes and their interaction, and they are expected to play a relevant role in the theory of periodic anomalous (rogue) waves (AWs) in nature. It is therefore important to establish the stability properties of these solutions under perturbations, to understand if they appear in nature, and in which form. It is rather well established that they are unstable with respect to small perturbations of the NLS equation. Concerning perturbations of these solutions within the NLS dynamics, there is the following common believe in the literature. Let the NLS background be unstable with respect to the first N modes; then i) if the M unstable modes of the AB M solution are strictly contained in this set (M < N ), then the AB M is unstable; ii) if they coincide with this set (M = N ), the so-called "saturation of the instability", then the AB M solution is neutrally stable. In this paper we argue instead that the AB M solution is always unstable, even in the saturation case M = N , and we prove it in the simplest case M = N = 1. We first prove the linear instability, constructing

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Section: Conclusion Open Problems and Remarksmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The linear and nonlinear instability of the Akhmediev breather

Grinevich,

Santini

2020

Preprint

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“…In the simplest case of one unstable mode only, this theory describes quantitatively a Fermi-Pasta-Ulam-Tsingou recurrence of AWs described by the Akhmediev breather (AB) [33,35]. In addition, a finitegap perturbation theory for 1+1 dimensional AWs has been also developed [17,18] to describe analytically the order one effect of small physical perturbations of the NLS model on the AW dynamics. See also [19].…”

Section: Introductionmentioning

confidence: 99%

The finite gap method and the periodic Cauchy problem of $2+1$ dimensional anomalous waves for the focusing Davey-Stewartson 2 equation

Grinevich¹,

Santini²

2022

Preprint

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The focusing Nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of 1 + 1 dimensional quasi monochromatic waves in weakly nonlinear media, and MI is considered the main physical mechanism for the appearence of anomalous (rogue) waves (AWs) in nature. In analogy with the recently developed analytic theory of periodic AWs of the focusing NLS equation, in this paper we extend these results to a 2 + 1 dimensional context, concentrating on the focusing Davey-Stewartson 2 (DS2) equation, an integrable 2+1 dimensional generalization of the focusing NLS equation. More precisely, we use the finite gap theory to solve, to leading order, the doubly periodic Cauchy problem of the focusing DS2 equation for small initial perturbations of the unstable background solution, what we call the periodic Cauchy problem of the AWs. As in the NLS case, we show that, to leading order, the solution of this Cauchy problem is expressed in terms of elementary functions of the initial data.

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“…However, the fluid dynamic significance still remains to be examined. We intended to scrutinize the relevance of rogue events in internal waves, the numerical robustness of triad rogue modes, as well as the connection with the classical problem of Fermi-Pasta-Ulam-Tsingou recurrence (FPUT) [21,22]. FPUT refers to the property or a tendency of a multi-mode nonlinear system to return to the initial states after complex stages of evolution.…”

Section: Introductionmentioning

confidence: 99%

Triads and Rogue Events for Internal Waves in Stratified Fluids with a Constant Buoyancy Frequency

Pan¹,

Yin²,

Chow³

2021

JMSE

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Internal waves in a stratified fluid with a constant buoyancy frequency were studied, with special attention given to rogue modes, extreme waves, dynamical evolution, and Fermi–Pasta–Ulam–Tsingou type recurrence phenomena. Rogue waves for triads in a general physical setting have recently been derived analytically, but the implications in a fluid mechanics context have not yet been fully assessed. Numerical simulations were conducted for cases of coupled triads where the common member is a daughter wave mode. In sharp contrast with previous studies, rogue modes instead of plane waves were used as the initial condition. Furthermore, spatial dependence was incorporated. Rogue or extreme waves in one set of triads provided a possible mechanism for significant energy transfer among modes of the internal wave spectrum, in addition to the other known theories, e.g., weak turbulence. Remarkably, Fermi–Pasta–Ulam–Tsingou recurrence types of growth and decay cycles arose, similarly to those observed for surface gravity wave groups governed by the cubic nonlinear Schrödinger equation. These mechanisms will enhance our understanding of transport processes in oceans.

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Fermi-Pasta-Ulam-Tsingou recurrence of periodic anomalous waves in the complex Ginzburg-Landau and in the Lugiato-Lefever equations

Cited by 15 publications

References 52 publications

The linear and nonlinear instability of the Akhmediev breather

The linear and nonlinear instability of the Akhmediev breather

The finite gap method and the periodic Cauchy problem of $2+1$ dimensional anomalous waves for the focusing Davey-Stewartson 2 equation

Triads and Rogue Events for Internal Waves in Stratified Fluids with a Constant Buoyancy Frequency

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