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

Abstract

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...

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

“…(1), with intricate rogue wave structures that resemble those displayed in the two-wave resonance case [25]. Indeed, when one field vanishes, these solutions can be readily reduced to those presented in Ref.…”

“…Indeed, when one field vanishes, these solutions can be readily reduced to those presented in Ref. [25]. In most cases, m and n are implicitly determined by the system of coupled algebraic equations (7) and (8), and thus one needs to solve for them numerically in order to find all possible rogue wave states.…”

“…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].…”

“…Here we exploited the Hirota bilinear method, which is expedient and efficient when the trial solutions can be easily deduced from the known solutions of a less complicated system [25]. After a little algebra, we obtain the exact fundamental rogue wave solutions of Eq.…”