Rogue waves, rational solitons, and modulational instability in an integrable fifth-order nonlinear Schrödinger equation

Yang, Yunqing; Yan, Zhenya; Malomed, Boris A.

doi:10.1063/1.4931594

Cited by 83 publications

(39 citation statements)

References 52 publications

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“…Various solutions of the QNLSE, such as solitons [48,49], breathers [49,50], and RWs [51], have been discussed in the literature. It is well known that breathers and solitons of arbitrary order can be obtained analytically using the DT technique.…”

Section: Dynamical Double-periodic Solutions Of the Extended Nlsementioning

confidence: 99%

Talbot carpets by rogue waves of extended nonlinear Schrödinger equations

et al. 2019

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We present analytical and numerical doubleperiodic solutions of the one-dimensional nonlinear Schrödinger equation and its extended versions in the form of Talbot carpets. The breathers and rogue waves of different orders are obtained using numerical simulations, starting from the initial conditions calculated by the Darboux transformation. To suppress undesirable aspects of modulation instability leading to homoclinic chaos, Fourier mode pruning procedures are invented to preserve and maintain the twofold periodicity of carpets. The novelty of this paper is analytical Talbot carpets for Hirota-quintic equation and ability to obtain them dynamically by controlling the growth of the Fourier modes. In addition, the new period-matching procedure is also described for periodic rogue waves that can be utilized to produce Talbot carpets without

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Section: Dynamical Double-periodic Solutions Of the Extended Nlsementioning

confidence: 99%

Talbot carpets by rogue waves of extended nonlinear Schrödinger equations

et al. 2019

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“…Representative key works, refer to third order (including modified Hirota [18], and Dysthe [19], equations) [20], [21], [22], as well as, fourth [23], [24], and fifth order [25], models. Important extensions to coupled equations and systems include [26] for parity-time symmetric systems, [27] for NLS systems with derivative nonlinearities, and [28] for Manakov systems (physically relevant in the context of Bose-Einstein condensates).…”

Section: Introductionmentioning

confidence: 99%

Excitation of Peregrine-Type Waveforms from Vanishing Initial Conditions in the Presence of Periodic Forcing

Karachalios

Kyriazopoulos

2019

Zeitschrift Für Naturforschung A

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We show by direct numerical simulations that spatiotemporally localized wave forms, strongly reminiscent of the Peregrine rogue wave, can be excited by vanishing initial conditions for the periodically driven nonlinear Schrödinger equation. The emergence of the Peregrine-type waveforms can be potentially justified, in terms of the existence and modulational instability of spatially homogeneous solutions of the model, and the continuous dependence of the localized initial data for small time intervals. We also comment on the persistence of the above dynamics, under the presence of small damping effects, and justify, that this behavior should be considered as far from approximations of the corresponding integrable limit.

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“…In selected special cases it was found [26][27][28][29][30], that rational solutions can still be identified, in Hirotatype variants of the original NLS equation (see also Ref. [31] for relevant work in higher-order NLS models).…”

Section: Introductionmentioning

confidence: 99%

Spatiotemporal algebraically localized waveforms for a nonlinear Schrödinger model with gain and loss

Anastassi

Fotopoulos

Frantzeskakis

et al. 2017

Physica D: Nonlinear Phenomena

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We consider the asymptotic behavior of the solutions of a nonlinear Schrödinger (NLS) model incorporating linear and nonlinear gain/loss. First, we describe analytically the dynamical regimes (depending on the gain/loss strengths), for finite-time collapse, decay, and global existence of solutions in the dynamics. Then, for all the above parametric regimes, we use direct numerical simulations to study the dynamics corresponding to algebraically decaying initial data. We identify crucial differences between the dynamics of vanishing initial conditions, and those converging to a finite constant background: in the former (latter) case we find strong (weak) collapse or decay, when the gain/loss parameters are selected from the relevant regimes. One of our main results, is that in all the above regimes, non-vanishing initial data transition through spatiotemporal, algebraically decaying waveforms. While the system is nonintegrable, the evolution of these waveforms is reminiscent to the evolution of the Peregrine rogue wave of the integrable NLS limit. The parametric range of gain and loss for which this phenomenology persists is also touched upon.

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Rogue waves, rational solitons, and modulational instability in an integrable fifth-order nonlinear Schrödinger equation

Cited by 83 publications

References 52 publications

Talbot carpets by rogue waves of extended nonlinear Schrödinger equations

Talbot carpets by rogue waves of extended nonlinear Schrödinger equations

Excitation of Peregrine-Type Waveforms from Vanishing Initial Conditions in the Presence of Periodic Forcing

Spatiotemporal algebraically localized waveforms for a nonlinear Schrödinger model with gain and loss

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