Rogue Wave Modes for the Long Wave–Short Wave Resonance Model

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

doi:10.7566/jpsj.82.074001

Cited by 56 publications

(19 citation statements)

References 29 publications

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“…Recently, a mathematical derivation of similar analytical rogue-wave solutions of the LWSW equation was also obtained by Chow et al using the Hirota bilinear method [34]. …”

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confidence: 99%

Dark- and bright-rogue-wave solutions for media with long-wave–short-wave resonance

2014

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Exact explicit rogue-wave solutions of intricate structures are presented for the long-wave-short-wave resonance equation. These vector parametric solutions feature coupled dark-and bright-field counterparts of the Peregrine soliton. Numerical simulations show the robustness of dark and bright rogue waves in spite of the onset of modulational instability. Dark fields originate from the complex interplay between anomalous dispersion and the nonlinearity driven by the coupled long wave. This unusual mechanism, not available in scalar nonlinear wave equation models, can provide a route to the experimental realization of dark rogue waves in, for instance, negative index media or with capillary-gravity waves. Because of their dramatic and potentially devastating manifestations, oceanic rogue waves have been the focus of intense research for more than a decade [1,2]. The roguewave terminology itself refers to the adverse surprise that is experienced when a transient giant wave of extreme amplitude or steepness is suddenly formed in the vicinity of a cruising ship [3]. In addition to being deployed into the safer and controllable environment of water tanks [4,5], rogue-wave research is also spreading widely to other disciplines that share general features of nonlinearity and complexity. These new avenues for rogue-wave research include fluid dynamics [6][7][8], nonlinear optics and lasers [9][10][11][12], plasma physics [13], and Bose-Einstein condensation [14]. The possibility to accede to a general understanding of rogue-wave formation is still an open question [15]. Nonetheless, the debate stimulates the comparison of predictions and observations between distinct areas, such as hydrodynamics and nonlinear optics, in situations where analogous dynamics can be identified through a common equation model. So far, the nonlinear Schrödinger (NLS) equation has played such a pivotal role.The Peregrine soliton, predicted 30 years ago [16], is the simplest rogue-wave solution associated with the NLS equation and it has recently been observed experimentally in a water-wave tank [4], a multicomponent plasma [13], and an optical fiber [10]. It is of fundamental significance because it is robust [17] and serves as the prototypical rogue-wave profile in various experimental fields. Indeed, the Peregrine soliton, as opposed to ordinary solitons that can be traced over long propagation distances, is localized in both transverse and propagation dimensions, thus reflecting the seemingly unpredictable appearance of a rogue wave [4,18]. Mathematically, the Peregrine soliton and related rogue-wave solutions of higher-order [5] While rogue-wave investigation is flourishing in several fields of science, there is a necessity to go beyond the standard NLS description in order to model important classes of physical systems in a relevant way. One recent development consists in including dissipative terms since a substantial supply of energy-from the wind in oceanography to the pump for laser cavities [19]-is generally required to drive roguew...

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“…Recently, a mathematical derivation of similar analytical rogue-wave solutions of the LWSW equation was also obtained by Chow et al using the Hirota bilinear method [34]. …”

mentioning

confidence: 99%

Dark- and bright-rogue-wave solutions for media with long-wave–short-wave resonance

2014

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“…H * is the conjugate function of H , G * is the conjugate function of G. Using the extended homoclinic test approach [21] we suppose…”

Section: Hirota's Bilinear Form and Rogue Wave Solutionsmentioning

confidence: 99%

“…have become a hot topic in many research fields [1][2][3][4]. Rogue waves were first observed in deep ocean [5].…”

Section: Introductionmentioning

confidence: 99%

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Rational homoclinic solution and rogue wave solution for the coupled long-wave–short-wave system

Chen

Dai

2015

Pramana - J Phys

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“…28 This bilinear method will also be shown here to be effective in the calculation of rogue waves. 29 It will be instructive to elucidate the contrast with the intensively studied integrable coupled NLS or Manakov equations. In terms of theoretical techniques, both the Darboux transformation 30 and the Hirota bilinear method 31 have been applied for the Manakov case, but only the Darboux technique has been utilized for the AB system.…”

Section: Formulationmentioning

confidence: 99%

A coupled “AB” system: Rogue waves and modulation instabilities

Grimshaw

Chow

et al. 2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Rogue waves are unexpectedly large and localized displacements from an equilibrium position or an otherwise calm background. For the nonlinear Schrödinger (NLS) model widely used in fluid mechanics and optics, these waves can occur only when dispersion and nonlinearity are of the same sign, a regime of modulation instability. For coupled NLS equations, rogue waves will arise even if dispersion and nonlinearity are of opposite signs in each component as new regimes of modulation instability will appear in the coupled system. The same phenomenon will be demonstrated here for a coupled "AB" system, a wave-current interaction model describing baroclinic instability processes in geophysical flows. Indeed, the onset of modulation instability correlates precisely with the existence criterion for rogue waves for this system. Transitions from "elevation" rogue waves to "depression" rogue waves are elucidated analytically. The dispersion relation as a polynomial of the fourth order may possess double pairs of complex roots, leading to multiple configurations of rogue waves for a given set of input parameters. For special parameter regimes, the dispersion relation reduces to a cubic polynomial, allowing the existence criterion for rogue waves to be computed explicitly. Numerical tests correlating modulation instability and evolution of rogue waves were conducted.

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Rogue Wave Modes for the Long Wave–Short Wave Resonance Model

Cited by 56 publications

References 29 publications

Dark- and bright-rogue-wave solutions for media with long-wave–short-wave resonance

Dark- and bright-rogue-wave solutions for media with long-wave–short-wave resonance

Rational homoclinic solution and rogue wave solution for the coupled long-wave–short-wave system

A coupled “AB” system: Rogue waves and modulation instabilities

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