This review is dedicated to recent progress in the active field of rogue waves, with an emphasis on the analytical prediction of versatile rogue wave structures in scalar, vector, and multidimensional integrable nonlinear systems. We first give a brief outline of the historical background of the rogue wave research, including referring to relevant up-to-date experimental results. Then we present an in-depth discussion of the scalar rogue waves within two different integrable frameworks-the infinite nonlinear Schrödinger (NLS) hierarchy and the general cubic-quintic NLS equation, considering both the self-focusing and self-defocusing Kerr nonlinearities. We highlight the concept of chirped Peregrine solitons, the baseband modulation instability as an origin of rogue waves, and the relation between integrable turbulence and rogue waves, each with illuminating examples confirmed by numerical simulations. Later, we recur to the vector rogue waves in diverse coupled multicomponent systems such as the long-wave short-wave equations, the three-wave resonant interaction equations, and the vector NLS equations (alias Topical Review

We study the existence and properties of rogue-wave solutions in different nonlinear wave evolution models that are commonly used in optics and hydrodynamics. In particular, we consider the Fokas-Lenells equation, the defocusing vector nonlinear Schrödinger equation, and the long-wave-short-wave resonance equation. We show that rogue-wave solutions in all of these models exist in the subset of parameters where modulation instability is present if and only if the unstable sideband spectrum also contains cw or zero-frequency perturbations as a limiting case (baseband instability). We numerically confirm that rogue waves may only be excited from a weakly perturbed cw whenever the baseband instability is present. Conversely, modulation instability leads to nonlinear periodic oscillations.

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We employ a nonrecursive Darboux transformation formalism for obtaining a hierarchy of rogue wave solutions to the focusing vector nonlinear Schrödinger equations (Manakov system). The exact explicit rogue wave solutions up to the second order are presented. Typical rogue wave patterns such as Peregrine-type, triple, quadruple, and sextuple vector rogue waves, either bright-dark or bright-bright in their respective components, are put forward. Despite the diversity, there exists a universal compossibility that different rogue wave states could coexist for the same background parameters. It is also shown that the higher-order rogue wave hierarchy can indeed be thought of as a nonlinear superposition of a fixed well prescribed number of fundamental rogue waves. These results may help understand the protean rogue wave manifestations in areas ranging from hydrodynamics to nonlinear optics.

Exact explicit rogue wave solutions of the Sasa-Satsuma equation are obtained by use of a Darboux transformation. In addition to the double-peak structure and an analog of the Peregrine soliton, the rogue wave can exhibit an intriguing twisted rogue-wave pair that involves four well-defined zero-amplitude points. This exotic structure may enrich our understanding on the nature of rogue waves.

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...

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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