On the Stabilization of Breather-type Solutions of the Damped Higher Order Nonlinear Schrödinger Equation

Schober, C. M.; Islas, A. L.

doi:10.3389/fphy.2021.633890

Cited by 5 publications

(6 citation statements)

References 38 publications

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“…(1) ǫ,β (x, t) in the two UM regime was not observed in the linear damped HONLS study [22] (for the same choice of T 0 in the initial data).…”

Section: Characterization Of Umentioning

confidence: 73%

“…Further, a weak instability has been shown to be associated with complex critical points which arise from a transverse intersection of bands of spectrum. In the waveform the instability is related to the mode switching from stationary to left or right traveling [22]. Numerical experiments show that the cumulative effect of instabilities arising from complex transverse critical points can be significant and the exact nature of this instability is under further investigation [22].…”

Section: Floquet Spectral Characterization Of Instabilitiesmentioning

confidence: 99%

“…It is well known that exponential instabilities are associated with complex "double points" and play a role in generating complex behavior in perturbations of the NLS equation [13,16]. Recently, in the unrestricted solution space, complex critical points arising from transverse intersections of bands of continuous spectrum were numerically shown to be associated with a weaker instability [22]. Section 2 provides background on the NLS spectral theory and it's use in distinguishing instabilities and the relevant nearby NLS states which are used to characterize the perturbed flow.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear damped spatially periodic breathers and the emergence of soliton-like rogue waves

Schober,

Islas

2022

Preprint

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The spatially periodic breather solutions (SPBs) of the nonlinear Schrödinger equation, prominent in modeling rogue waves, are unstable. In this paper we numerically investigate the effects of nonlinear dissipation and higher order nonlinearities on the routes to stability of the SPBs in the framework of the nonlinear damped higher order nonlinear Schrödinger (NLD-HONLS) equation. The initial data used in the experiments are generated by evaluating exact SPB solutions at time T 0 . The number of instabilities of the background Stokes wave and the damping strength are varied. The Floquet spectral theory of the NLS equation is used to interpret and provide a characterization of the perturbed dynamics in terms of nearby solutions of the NLS equation. Significantly, as T 0 is varied, tiny bands of complex spectrum are observed to pinch off in the Floquet decomposition of the NLD-HONLS data, reflecting the breakup of the SPB into a waveform that is close to either a one or two "soliton-like" structure. For wide ranges of T 0 , i.e. for solutions initialized in the early to middle stage of the development of the MI, all rogue waves are observed to occur when the spectrum is close to a one or two soliton-like state. When the solutions are initialized as the MI is saturating, rogue waves also can occur after the spectrum has left a soliton-like state. Other novel features arise due to nonlinear damping: enhanced asymmetry, two timescales in the evolution of the spectrum and a delay in the growth of instabilities due to frequency downshifting.

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“…(1) ǫ,β (x, t) in the two UM regime was not observed in the linear damped HONLS study [22] (for the same choice of T 0 in the initial data).…”

Section: Characterization Of Umentioning

confidence: 73%

Section: Floquet Spectral Characterization Of Instabilitiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nonlinear damped spatially periodic breathers and the emergence of soliton-like rogue waves

Schober,

Islas

2022

Preprint

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“…Our aim is to show that linear damping causes, instead, the re-currences to undergo a dynamical change from one type to the other, through a phenomenon of loss-induced separatrix crossing. Remarkably, we find by perturbative arguments (based either on a Fourier mode-truncation or finite-gap theory [46,47]) that such crossing occurs around multiple critical values of the linear damping coefficient. We demonstrate this by reporting evidence for the first two critical loss values in a fiber optics experiment, where we reconstruct, via non-destructive measurements, the nonlinear evolution of MI in power and phase, as the effective losses are accurately tailored via Raman amplification techniques [48][49][50].…”

mentioning

confidence: 82%

Multiple symmetry breaking induced by weak damping in the Fermi-Pasta-Ulam-Tsingou recurrence process

Vanderhaegen¹,

Szriftgiser²,

Kudlinski³

et al. 2022

Preprint

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We show that even small dissipation can strongly affect the Fermi-Pasta-Ulam-Tsingou recurrence phenomenon. Taking the fully nonlinear stage of modulational instability as an experimentally accessible example, we show that the linear attenuation induces the symmetry of the recurrence to be broken through separatrix crossing occurring at multiple critical values of the attenuation. We provide experimental evidence for this phenomenon in a fiber optics experiment designed in such a way that the effective losses can be carefully tailored by techniques based on Raman amplification.

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“…This solution is based on the proper adaptation of the finite-gap method [32,48,53,56,62,66] (see [52] for its first application to NLS) to it. See also [39] for an alternative approach to the study of the AW recurrence, based on matched asymptotic expansions; see [41] for the analytic study of the phase resonances in the AW recurrence; see [30,31,72] 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 [63,77], see [29,73] for the study of the stabilisation effects of higher-order corrections to NLS. The fundamental matrix solution T(λ, x, y, t) of ( 4)- (6) in the x-periodic problem, such that T(λ, y, y, t) = E, where E is the identity matrix (see [34]), is an entire function of λ.…”

Section: Introductionmentioning

confidence: 99%

The linear and nonlinear instability of the Akhmediev breather

Grinevich

Santini

2021

Nonlinearity

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The Akhmediev breather (AB) and its M-breather generalisation, hereafter called AB M , are exact solutions of the focusing NLS equation periodic in space and exponentially localised 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 in nature. It is therefore important to establish the stability properties of these solutions under perturbations. Concerning perturbations of these solutions within the NLS dynamics, there is the following common belief 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), then the AB M solution is neutrally stable. In this paper we argue instead that the AB M solution is always linearly unstable, even in the saturation case M = N, and we prove it in the simplest case M = N = 1, constructing two examples of x-periodic solutions of the linearised theory growing exponentially in time. Then we sketch the proof of completeness of the basis of periodic solutions of the linearised theory. We also investigate the nonlinear instability showing that (i) a perturbed AB initial condition evolves into a recurrence of ABs; (ii) the AB solution is more unstable than the background solution, and its instability increases as T → 0, where T is the AB appearance time. Although the AB solution is linearly and nonlinearly unstable, its instability generates a recurrence of ABs, and this recurrence implies its relevance in the natural phenomena described by the NLS equation, as well as its orbital stability, using a specific definition of orbital stability present in the literature.

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On the Stabilization of Breather-type Solutions of the Damped Higher Order Nonlinear Schrödinger Equation

Cited by 5 publications

References 38 publications

Nonlinear damped spatially periodic breathers and the emergence of soliton-like rogue waves

Nonlinear damped spatially periodic breathers and the emergence of soliton-like rogue waves

Multiple symmetry breaking induced by weak damping in the Fermi-Pasta-Ulam-Tsingou recurrence process

The linear and nonlinear instability of the Akhmediev breather

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