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

Abstract

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...

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

“…However, both modes here share the same baseband growth rate because they correspond to a pair of complex conjugate roots of Equation (3). Similar co-existence of rogue waves in a chaotic wave field was also reported earlier in the literature [40]. …”

“…From Equation (3), if a + ib is a root of the dispersion relation, then a − ib will also be admissible and provides another rogue wave solution. This phenomenon was also observed in other multi-component system [38][39][40][41]. Such multi-rogue-wave scenarios are not allowed in the two-component Manakov system.…”

Rogue waves for a system of coupled derivative nonlinear Schrödinger equations

“…However, the wave patterns associated with the first pair are drastically different from those corresponding to the second pair. Indeed, multiple roots of dispersion relations of RW systems, e.g., long-wave-short-wave interaction model [30], are known to produce different wave profiles.…”

Rogue Waves for an Alternative System of Coupled Hirota Equations: Structural Robustness and Modulation Instabilities

“…While similar approaches might have been adopted in earlier works , we also exhibit the connection between the instability gain spectrum and RWs by following the time evolution of a plane wave subjected to a disturbance of a specified frequency. With information from the MI gain spectrum, further quantitative trends can be deduced.…”