On multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equation

Dubard, Philippe; Gaillard, Pierre; Klein, Christian; Matveev, V. I.

doi:10.1140/epjst/e2010-01252-9

Cited by 277 publications

(243 citation statements)

References 11 publications

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“…The latter behavior could be linked to the existence of high-order multiple rogue wave solutions of Eq. (1), in analogy to the multiple rogue wave solutions found within the scalar NLS equation [46][47][48]. These subsequent multiple rogue waves could also be triggered by the onset of MI, which promotes quasiperiodic structures by patterning the continuous-wave background.…”

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confidence: 87%

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

2014

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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...

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Section: Fig 2 (Color Online)mentioning

confidence: 87%

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

2014

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“…The richer family of multirogue wave solutions of the NLSE is also very attractive for future works. These more general higher-order solutions exhibit multiple maxima, i.e., multiple rogue waves, with a complex spatiotemporal arrangement [27,[33][34][35][36].…”

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confidence: 99%

Collision of Akhmediev Breathers in Nonlinear Fiber Optics

2013

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We report here a novel fiber-based test bed using tailored spectral shaping of an optical-frequency comb to excite the formation of two Akhmediev breathers that collide during propagation. We have found specific initial conditions by controlling the phase and velocity differences between breathers that lead, with certainty, to their efficient collision and the appearance of a giant-amplitude wave. Temporal and spectral characteristics of the collision dynamics are in agreement with the corresponding analytical solution. We anticipate that experimental evidence of breather-collision dynamics is of fundamental importance in the understanding of extreme ocean waves and in other disciplines driven by the continuous nonlinear Schrödinger equation.

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“…Finally, we would like to study the nonlinear interplay characteristic of the wave transition. It is well-known that the higher-order rogue wave exhibits rich structure characteristic [31][32][33][34][35][36][37]. Thus, it is necessary to answer how about the property of the higher-order wave transition.…”

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confidence: 99%

State transition induced by higher-order effects and background frequency

et al. 2015

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State transition between the Peregrine rogue wave and w-shaped traveling wave induced by higherorder effects and background frequency is studied. We find that this intriguing transition, described by an exact explicit rational solution, is consistent with the modulation instability (MI) analysis that involves MI region and stability region in low perturbation frequency region. In particular, the link between the MI growth rate and transition characteristic analytically demonstrates that, the size characteristic of transition is positively associated with the reciprocal of zero-frequency growth rate. Further, we investigate the case for nonlinear interplay of multi-localized waves. It is interesting that the interaction of second-order waves in stability region features a line structure, rather than an elastic interaction between two w-shaped traveling waves. [7,8] of weakly modulated plane wave. More specifically, rogue waves exist only in the MI subregion where the instability frequency band is close to the vanishing frequency [9]. In this regard, rogue wave has a consistent structure in the standard nonlinear Schrödinger equation (NLSE) where the MI characteristic remains invariant in nature. Nevertheless, the MI often exhibits some interesting features when the additional physical effects are taken into consideration, such as cross-phase modulation [10, 11], higher-order perturbation terms [12], etc. Thus, it is important to study the rogue wave property induced by the features of MI growth rate distribution.Recent studies demonstrate that rogue wave can exhibit structural diversity beyond the reach of the standard NLSE in presence of higher-order effects [13][14][15][16][17][18][19][20][21][22][23][24]. However, to our knowledge, less attention has been paid to analyzing rogue wave property in combination with the distribution characteristic of corresponding MI growth rate. Here we find that, with some higher-order perturbation terms (the third-order dispersion and delayed nonlinear response term), the MI growth rate shows a non-uniform distribution characteristic in low perturbation frequency region, in particular, it opens up a stability region as background frequency changes (see Fig. 1). Thus we anticipate that there will be some interesting physical properties as a rogue wave evolves and approaches the stability region.As a starting point, we address the problem via a completely integrable higher-order NLSE-the Hirota equa- * Electronic address: zyyang@nwu.edu.cn † Electronic address: zhaolichen3@163.com tion (HE) [25]-which involves the higher-order perturbation effects. In dimensionless form, the HE readswhere z is the propagation variable, t is the retarded time in a moving frame with the group velocity, and E(t, z) is the slowly varying envelope of the wave field. The real parameter β is introduced to be responsible for the thirdorder dispersion and delayed nonlinear response term, respectively. If β = 0, it reduces to the standard NLSE. The existence of rogue waves in the HE has been recently dem...

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On multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equation

Cited by 277 publications

References 11 publications

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

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

Collision of Akhmediev Breathers in Nonlinear Fiber Optics

State transition induced by higher-order effects and background frequency

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