Integrability and Linear Stability of Nonlinear Waves

Degasperis, A.; Lombardo, Sara; Sommacal, Matteo

doi:10.1007/s00332-018-9450-5

Cited by 26 publications

(44 citation statements)

References 54 publications

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“…We remark that the Peregrine solitons are homoclinic, describing AWs appearing apparently from nowhere and desappearing in the future, while the multisoliton solutions of Akhmediev type are almost homoclinic, returning to 3 the original background up to a multiplicative phase factor. Generalizations of these solutions to the case of integrable multicomponent NLS equations have also been found [18,34,35].…”

Section: Introductionmentioning

confidence: 85%

The finite-gap method and the periodic NLS Cauchy problem of anomalous waves for a finite number of unstable modes

Grinevich¹,

Santini²

2019

Russ. Math. Surv.

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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, and MI is considered the main physical mechanism for the appearence of anomalous (rogue) waves (AWs) in nature. In this paper we study, using the finite gap method, the NLS Cauchy problem for generic periodic initial perturbations of the unstable background solution of NLS (what we call the Cauchy problem of the AWs), in the case of a finite number N of unstable modes. We show how the finite gap method adapts to this specific Cauchy problem through three basic simplifications, allowing one to construct the solution, at the leading and relevant order, in terms of elementary functions of the initial data. More precisely, we show that, at the leading order, i) the initial data generate a partition of the time axis into a sequence of finite intervals; ii) in each interval I of the partition, only a subset of N (I) ≤ N unstable modes are "visible", and iii) the NLS solution is approximated, for t ∈ I, by the N (I)-soliton solution of Akhmediev type, describing the nonlinear interaction of these visible unstable modes, whose parameters are expressed in terms of the initial data through elementary functions. This result explains the relevance of the m-soliton solutions of Akhmediev type, with m ≤ N , in the generic periodic Cauchy problem of the AWs, in the case of a finite number N of unstable modes. 1 arXiv:1810.09247v2 [math-ph]

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Section: Introductionmentioning

confidence: 85%

The finite-gap method and the periodic NLS Cauchy problem of anomalous waves for a finite number of unstable modes

Grinevich¹,

Santini²

2019

Russ. Math. Surv.

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“…Since a main ingredient in the occurrence of rogue waves has been recognised [38,39] to be the instability of their background state, our main focus here is on the linear stability of continuous waves (CW). Moreover, it has been also realised [46] that the stability of two CWs, which are weakly at resonance, features a richer phenomenology with respect to that associated to just one plane wave [47].…”

Section: Cnls: Spectra and Stabilitymentioning

confidence: 99%

“…It has been used in several contexts and in different formulations, in particular in soliton perturbation theory, where these eigenmodes have been termed squared eigenfunctions [52,53]. A general derivation of Equation (10) from the Lax pair is given in [46], with no need to cope with the burden of the direct and inverse spectral problem techniques.…”

Section: Cnls: Spectra and Stabilitymentioning

confidence: 99%

“…are conveniently replacing the coupling constants s 1 and s 2 and the amplitudes a 1 and a 2 . As for the parameter q, it is worth observing that the case q = 0 can be explicitly treated [46], and it leads to defocusing and focusing effects, which are similar to those associated to the NLS equation. Thus, we consider only non-vanishing values of q, say q = 0, which could also be rescaled to the value q = 1.…”

Section: Cnls: Spectra and Stabilitymentioning

confidence: 99%

“…Thus, we consider only non-vanishing values of q, say q = 0, which could also be rescaled to the value q = 1. We also observe that one may consider, with no loss of generality (see [46]), only non-negative values of r and positive values of q. Thus, the parameter space we explore in the following analysis is the half (r, p) plane with r ≥ 0.…”

Section: Cnls: Spectra and Stabilitymentioning

confidence: 99%

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Rogue Wave Type Solutions and Spectra of Coupled Nonlinear Schrödinger Equations

Degasperis

Lombardo

Sommacal

2019

Fluids

Self Cite

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The formation of rogue oceanic waves may be the result of different causes. Various factors (winds, currents, dispersive focussing, depth, nonlinear focussing and instability) make this subject intriguing, and yet its understanding is quite relevant to practical issues. Here, we deal only with the nonlinear character of this dynamics, which has been recognised as the main ingredient to rogue wave formation. In this perspective, the formation of rogue waves requires a non-vanishing and unstable background such as a nonlinear regular wave train with attractive self-interaction. The simplest, best known model of such dynamics is the universal nonlinear Schrödinger equation. This has proven to serve as a good approximation in various contexts and over a broad range of experimental settings. This model aims to give the slow evolution of the envelope of one monochromatic wave due to nonlinearity. Here, we naturally consider the same problem for the envelopes of two weakly resonant monochromatic waves. As for the nonlinear Schrödinger equation, which is integrable, we adopt an integrable model to describe the interaction of two waves. This is the system of two coupled nonlinear Schrödinger equations (Manakov model) with self- and cross-interactions that may be both defocussing and focussing. We first discuss the linear stability properties of the background by computing the spectrum for all values of the parameters such as coupling constants and amplitudes. In particular, we relate the instability bands to properties of the spectrum and compute the gain function (or growth rate). We also relate to the stability spectrum the value of the spectral variable, which corresponds to a rogue wave solution. In contrast with the nonlinear Schrödinger equation, different types of single rogue wave exist that correspond to different values of the spectral variable even in the same spectrum. For these critical values, which are completely classified, we give the corresponding explicit expression of the rogue wave solution that follows from the well known Darboux–Dressing transformation method. Although not all systems of two coupled nonlinear Schrödinger equations that have been derived in water wave dynamics are integrable, our investigation contributes to the understanding of new effects due to wave coupling, at least for model equations that, even if not integrable, are close enough to the model considered here. For instance, our findings lead to investigate rogue waves generated by instabilities due to self- and cross-interactions of defocusing type. An illustrative selection of two coupled rogue waves solutions is displayed.

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