Rogue wave modes for a derivative nonlinear Schrödinger model

Chan, Hiu Ning; Chow, K. W.; Kedziora, David J.; Grimshaw, Roger; Ding, Edwin

doi:10.1103/physreve.89.032914

Cited by 96 publications

(64 citation statements)

References 30 publications

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“…A similar scenario prevails in the present DNLSE system. For the single component DNLSE, leading order RWs in the form of elevation patterns were documented earlier [25]. For coupled DNLSEs, 17 depression and four-petal patterns appear, see Figs.…”

Section: Variation In Wave Profilesmentioning

confidence: 88%

“…For the NLSE, the amplification ratio (the largest displacement featured by a RW solution divided by that of the background) for the Peregrine breather is 3 [5,6], which also holds for the first-order RW solutions of the DNLSE [25]. For the integrable Manakov system, the amplification ratio of first order RWs without group velocity mismatch cannot exceed three [10,11,13].…”

Section: Wave Profiles and Dynamicsmentioning

confidence: 99%

“…We shall employ the Hirota bilinear transform, as this technique has been shown to work effectively for calculating multi-soliton solutions [23,24], as well as RWs [25] of integrable nonlinear evolution equations. This method may be utilized as an alternative to the widely used Darboux transformation [5].…”

Section: Introductionmentioning

confidence: 99%

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Rogue waves for a system of coupled derivative nonlinear Schrödinger equations

et al. 2016

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Rogue waves (RWs) are unexpectedly strong excitations emerging from an otherwise tranquil background. The nonlinear Schrödinger equation (NLSE), a ubiquitous model with wide applications to fluid mechanics, optics and plasmas, exhibits RWs only in the regime of modulation instability (MI) of the background.For system of multiple waveguides, the governing coupled NLSEs can produce regimes of MI and RWs, even if each component has dispersion and cubic nonlinearity of opposite signs. A similar effect will be demonstrated for a system of coupled derivative NLSEs (DNLSEs), where the special feature is the nonlinear self-steepening of narrow pulses. More precisely, these additional regimes of MI and RWs for coupled DNLSEs will depend on the mismatch in group velocities between the components, as well as the parameters for cubic nonlinearity and self-steepening. RWs considered in this work differ from those of the NLSEs in terms of the amplification ratio and criteria of existence.Applications to optics and plasma physics are discussed.3

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Section: Variation In Wave Profilesmentioning

confidence: 88%

Section: Wave Profiles and Dynamicsmentioning

confidence: 99%

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Rogue waves for a system of coupled derivative nonlinear Schrödinger equations

et al. 2016

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“…The Peregrine breather of the NLS equation is localized in both space and time, and is a widely utilized model for rogue waves [14,15]. This solution is only nonsingular in the focusing regime unless higher order terms are considered [16].…”

Section: Introductionmentioning

confidence: 99%

Rogue Wave Modes for the Coupled Nonlinear Schrödinger System with Three Components: A Computational Study

Chan¹,

Chow²

2017

Applied Sciences

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Abstract:The system of "integrable" coupled nonlinear Schrödinger equations (Manakov system) with three components in the defocusing regime is considered. Rogue wave solutions exist for a restricted range of group velocity mismatch, and the existence condition correlates precisely with the onset of baseband modulation instability. This assertion is further elucidated numerically by evidence based on the generation of rogue waves by a single mode disturbance with a small frequency. This same computational approach can be adopted to study coupled nonlinear Schrödinger equations for the "non-integrable" regime, where the coefficients of self-phase modulation and cross-phase modulation are different from each other. Starting with a wavy disturbance of a finite frequency corresponding to the large modulation instability growth rate, a breather can be generated. The breather can be symmetric or asymmetric depending on the magnitude of the growth rate. Under the presence of a third mode, rogue wave can exist under a larger group velocity mismatch between the components as compared to the two-component system. Furthermore, the nonlinear coupling can enhance the maximum amplitude of the rogue wave modes and bright four-petal configuration can be observed.

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“…Naturally, the higher order RW solutions have more peaks and exhibit several interesting patterns [39][40][41][42][43][44]. In addition to the NLS equation, there are many other equations admitting RW (or Peregrine-type) solutions such as the modified Korteweg-de Vries equation, the Fokas-Lenells equation, the derivative NLS equation, the long-wave-short-wave resonance equation, the vector NLS, the Davey-Stewartson equation and the KP-I equation [45][46][47][48][49][50][51][52][53][54][55][56][57][58][59], etc.…”

Section: Indiamentioning

confidence: 99%

The Darboux transformation of the Kundu–Eckhaus equation

Qiu

Zhang³

et al. 2015

Proc. R. Soc. A.

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We construct an analytical and explicit representation of the Darboux transformation (DT) for the KunduEckhaus (KE) equation. Such solution and n-fold DT T n are given in terms of determinants whose entries are expressed by the initial eigenfunctions and 'seed' solutions. Furthermore, the formulae for the higher order rogue wave (RW) solutions of the KE equation are also obtained by using the Taylor expansion with the use of degenerate eigenvalues. ., all these parameters will be defined latter. These solutions have a parameter β, which denotes the strength of the non-Kerr (quintic) nonlinear and the self-frequency shift effects. We apply the contour line method to obtain analytical formulae of the length and width for the first-order RW solution of the KE equation, and then use it to study the impact of the β on the RW solution. We observe two interesting results on localization characters of β, such that if β is increasing from a/2: (i) the length of the RW solution is increasing as well, but the width is decreasing; (ii) there exist a significant rotation of the RW along the clockwise direction. We also observe the oppositely varying trend if β is increasing to a/2. We define an area of the RW solution and find that this area associated with c = 1 is invariant when a and β are changing.

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Rogue wave modes for a derivative nonlinear Schrödinger model

Cited by 96 publications

References 30 publications

Rogue waves for a system of coupled derivative nonlinear Schrödinger equations

Rogue waves for a system of coupled derivative nonlinear Schrödinger equations

Rogue Wave Modes for the Coupled Nonlinear Schrödinger System with Three Components: A Computational Study

The Darboux transformation of the Kundu–Eckhaus equation

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