Rogue waves in the Davey-Stewartson I equation

Ohta, Yuichi; Yang, Jianke

doi:10.1103/physreve.86.036604

Cited by 341 publications

(224 citation statements)

References 32 publications

(60 reference statements)

Supporting

Mentioning

208

Contrasting

Unclassified

Order By: Relevance

“…When compared to scalar dynamical systems, vector systems generally allow energy transfer between their additional degrees of freedom, which potentially yields families of intricate vector rogue-wave solutions. Indeed, rogue-wave families with complicated rational forms have been recently found in the Davey-Stewartson equation [23], the coupled Manakov system [24], and the coupled Hirota equations [25]. Let us recall that the scalar NLS equation does not admit single dark-rogue-wave solutions, even in the case of a defocusing nonlinearity.…”

mentioning

confidence: 99%

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

2014

View full text Add to dashboard Cite

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...

show abstract

mentioning

confidence: 99%

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

2014

View full text Add to dashboard Cite

show abstract

“…Recent developments have taken into account dissipative effects [11,15,16], included higher-order nonlinear terms [17][18][19], or considered the coupling between several fields [20][21][22][23][24][25]. The latter investigations have led to the discovery of intricate rogue wave structures that are generally unattainable in the scalar models.…”

mentioning

confidence: 99%

Coexisting rogue waves within the (2+1)-component long-wave–short-wave resonance

2014

View full text Add to dashboard Cite

The coexistence of two different types of fundamental rogue waves is unveiled, based on the coupled equations describing the (2+1)-component long-wave-short-wave resonance. For a wide range of asymptotic background fields, each family of three rogue wave components can be triggered by using a slight deterministic alteration to the otherwise identical background field. The ability to trigger markedly different rogue wave profiles from similar initial conditions is confirmed by numerical simulations. This remarkable feature, which is absent in the scalar nonlinear Schrödinger equation, is attributed to the specific three-wave interaction process and may be universal for a variety of multicomponent wave dynamics spanning from oceanography to nonlinear optics. [4,5], these extreme wave events were also observed in a wide class of physical systems, including capillary waves and surface ripples [6,7], plasmas [8], optical fibers [9,10], mode-locked lasers [11], and filaments [12]. These studies have uncovered general features of nonlinearity and complexity shared by rogue waves, e.g., they are extremely large and steep compared with typical events, occur in a nonlinear medium, and follow an unusual L-shaped statistics [9,[11][12][13]. Despite these diverse features, mathematical solutions of rogue waves can be expressed as rational functions localized in both space and time. One typical example is the Peregrine soliton [14], a well-known rogue wave prototype in various experimental fields [4,8,10], which is the lowest-order rational solution to the nonlinear Schrödinger (NLS) equation.Following the need to model complex physical systems more precisely, it has become important to study rogue wave phenomena beyond the framework of the NLS equation. Recent developments have taken into account dissipative effects [11,15,16], included higher-order nonlinear terms [17][18][19], or considered the coupling between several fields [20][21][22][23][24][25]. The latter investigations have led to the discovery of intricate rogue wave structures that are generally unattainable in the scalar models. In particular, we showed in Ref.[25] that the long-wave-short-wave (LWSW) resonance interaction can result in stable dark and bright rogue waves in spite of the onset of modulational instability (MI). This finding brings about the possibility to observe dark rogue waves in LWSW resonance systems such as negative index media [26] and capillary-gravity waves [6,27].Basically, the LWSW resonance is a general parametric process that manifests when the group velocity of the short wave matches the phase velocity of the long wave [28]. It has been predicted in different disciplines such as fluid dynamics [27], plasma physics [29], oceanography [30], and nonlinear optics [26,31]. Early works showed that both the LWSW and the NLS equations could be obtained from the same Davey-Stewartson system under the appropriate parameter conditions [27,32,33], although the former is currently less popular in use than the latter.In fact, it is possible to co...

show abstract

“…However, the NLS equation includes the lowest-order dispersion and lowest-order nonlinearity. Therefore, in order to obtain some specific structures of the optical rogue wave, extended NLS equation including higher-order dispersions and nonlinearity have been attracting much more attentions, such as the Hirota equation [13], Sasa-Satsuma (SS) equation [14,15], Chen-Lee-Liu (CLL)-type and Kaup-Newell (KN)-type derivative NLS equations [16], GerdjikovIvanov (GI) equation [17], Lakshmanan-PorsezianDaniel (LPD) equation [18], the set of coupled NLS equtaion [6,[19][20][21][22], and so on [23]. At the same time, approximate solutions in the form of rogue wave can also be obtained in the equations which are not integrable [24].…”

Section: Introductionmentioning

confidence: 99%

Optical temporal rogue waves in the generalized inhomogeneous nonlinear Schrödinger equation with varying higher-order even and odd terms

2015

View full text Add to dashboard Cite

We investigate optical temporal rogue waves of the generalized nonlinear Schrödinger (NLS) equation with higher-order odd and even terms and space-and time-modulated coefficients, which includes the NLS equation, Lakshmanan-Porsezian-Daniel (LPD) equation, Hirota equation, Chen-Lee-Liu equation, and Kaup-Newell derivative NLS equation. Based on the similarity reduction method, the generalized NLS equation can be reduced to the integrable LPDHirota equation under a set of constraint conditions, from which the solutions of generalized NLS equation can be obtained in the basis of solutions of the LPDHirota equation and the similarity transformation. In particular, the first-and second-order self-similar rogue wave solutions of the generalized NLS equation are derived under different parameters, and the contour profiles and density evolutions of self-similar rogue wave solutions are given to study their wave structures and dynamic properties. At the same time, the motions of the hump and valleys related to the self-similar rogue waves are also given, from which we can control and manage the self-similar rogue waves by adjusting the third-order dispersion term. These results may be useful in nonlinear optics and related fields. KeywordsThe generalized nonlinear Schrödinger (NLS) equation · The LPD-Hirota equation · Similarity reduction method · Rogue wave solutions · Motions of the hump and valleys

show abstract

Rogue waves in the Davey-Stewartson I equation

Cited by 341 publications

References 32 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

Coexisting rogue waves within the (2+1)-component long-wave–short-wave resonance

Optical temporal rogue waves in the generalized inhomogeneous nonlinear Schrödinger equation with varying higher-order even and odd terms

Contact Info

Product

Resources

About