Numerical Instability of the Akhmediev Breather and a Finite-Gap Model of It

Grinevich, P. G.; Santini, P. M.

doi:10.1007/978-3-030-04807-5_2

Cited by 11 publications

(18 citation statements)

References 55 publications

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“…In the second subcase in which more than one unstable gap is open, we obtain the elementary description of the first nonlinear stage of MI, given again by the Akhmediev 1-breather solution, and how perturbations due to numerics and/or real experiments can affect this result. In addition, in [26] we have shown that the finite gap recurrence formulas constructed in [25] are a good model for describing the numerical (and physical) instabilities of the Akhmediev breather.…”

mentioning

confidence: 92%

The exact rogue wave recurrence in the NLS periodic setting via matched asymptotic expansions, for 1 and 2 unstable modes

Grinevich

Santini

2018

Physics Letters A

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The focusing Nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, the main physical mechanism for the generation of rogue (anomalous) waves (RWs) in Nature. In this paper we investigate the x-periodic Cauchy problem for NLS for a generic periodic initial perturbation of the unstable constant background solution, in the case of N = 1, 2 unstable modes. We use matched asymptotic expansion techniques to show that the solution of this problem describes an exact deterministic alternate recurrence of linear and nonlinear stages of MI, and that the nonlinear RW stages are described by the N-breather solution of Akhmediev type, whose parameters, different at each RW appearence, are always given in terms of the initial data through elementary functions. This paper is motivated by a preceeding work of the authors in which a different approach, the finite gap method, was used to investigate periodic Cauchy problems giving rise to RW recurrence.

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confidence: 92%

The exact rogue wave recurrence in the NLS periodic setting via matched asymptotic expansions, for 1 and 2 unstable modes

Grinevich

Santini

2018

Physics Letters A

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“…To conclude, the disagreement between the analytic breather solution of the NLS equation and its numerical simulation presented here (Fig. 2) can occur due to the following reasons (see also [28]): i) due to the different dynamical features of the continuous partial differential equation and its discrete counterpart;…”

Section: Simulations Of the Nonlinear Schrodinger Equationmentioning

confidence: 82%

“…The perturbations may be a numerical noise which cannot be removed completely, thus the situation when the excitation of one unstable mode leads to the development of all available unstable modes represents the general picture (see, e.g., in [12,27]). The impact of different numerical effects on the accuracy of the simulation of the Akhmediev breathers was also in the focus of the study [28].…”

Section: The Circumstances Which Can Prevent the Full Recurrence Of Tmentioning

confidence: 99%

“…Besides, some simulations were verified by means of the ultra-high precision calculations using arbitrary-precision arithmetic (with 256-bit long mantissa, which is roughly equivalent to 80 decimal digits) and with the iterations in time performed by the 4-5-order Runge-Kutta method. We should mention that the problem of stable numerical simulation of the Akhmediev breathers was also tackled in [28] in a different aspect.…”

Section: Simulations Of the Nonlinear Schrodinger Equationmentioning

confidence: 99%

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On the incomplete recurrence of modulationally unstable deep-water surface gravity waves

Slunyaev

Dosaev

2019

Communications in Nonlinear Science and Numerical Simulation

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The issue of a recurrence of the modulationally unstable water wave trains within the framework of the fully nonlinear potential Euler equations is addressed. It is examined, in particular, if a modulation which appears from nowhere (i.e., is infinitesimal initially) and generates a rogue wave which then disappears with no trace. If so, this wave solution would be a breather solution of the primitive hydrodynamic equations. It is shown with the help of the fully nonlinear numerical simulation that when a rogue wave occurs from a uniform Stokes wave train, it excites other waves which have different lengths, what prevents the complete recurrence and, eventually, results in a quasi-periodic breathing of the wave envelope. Meanwhile the discovered effects are rather small in magnitude, and the period of the modulation breathing may be thousands of the dominant wave periods. Thus, the obtained solution may be called a quasi-breather of the Euler equations.

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“…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 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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Numerical Instability of the Akhmediev Breather and a Finite-Gap Model of It

Cited by 11 publications

References 55 publications

The exact rogue wave recurrence in the NLS periodic setting via matched asymptotic expansions, for 1 and 2 unstable modes

The exact rogue wave recurrence in the NLS periodic setting via matched asymptotic expansions, for 1 and 2 unstable modes

On the incomplete recurrence of modulationally unstable deep-water surface gravity waves

The linear and nonlinear instability of the Akhmediev breather

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