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

: The system of “integrable” coupled nonlinear Schrödinger equations (Manakov system) with three components in the defocusing regime is considered. Rogue wave solutions exist for a restricted range of group velocity mismatch, and the existence condition correlates precisely with the onset of baseband modulation instability. This assertion is further elucidated numerically by evidence based on the generation of rogue waves by a single mode disturbance with a small frequency. This same computational approach can be adopted to study coupled nonlinear Schrödinger equations for the “non-integrable” regime, where the coefﬁcients of self-phase modulation and cross-phase modulation are different from each other. Starting with a wavy disturbance of a ﬁnite frequency corresponding to the large modulation instability growth rate, a breather can be generated. The breather can be symmetric or asymmetric depending on the magnitude of the growth rate. Under the presence of a third mode, rogue wave can exist under a larger group velocity mismatch between the components as compared to the two-component system. Furthermore, the nonlinear coupling can enhance the maximum amplitude of the rogue wave modes and bright four-petal conﬁguration can be observed.


Introduction
Rogue waves or freak waves are extreme events in the ocean which are characterized by the emergence of large waves from an otherwise tranquil background [1][2][3]. The unexpectedly large displacements of the sea surface pose threats to maritime activities and offshore structures. Intensive research efforts are conducted to understand the physics of rogue wave and to develop measures to predict or detect such waves [4,5]. Although rogue waves originate from the context of water waves [1][2][3][6][7][8][9], these studies have been extended to other physical contexts like optical fibers [10][11][12] and Bose-Einstein condensates [13]. Moreover, it has been demonstrated that optical rogue waves are related to supercontinuum generation [10,11]. The high repetition rate of optical experiments is an advantage in the study of these rare events.
For hydrodynamic surface waves, the nonlinear Schrödinger (NLS) equation governs the slow evolution of a weakly nonlinear wave packet [2]. The NLS equation can also describe the dynamics of temporal pulses in an optical fiber [10]. The Peregrine breather of the NLS equation is localized in both space and time, and is a widely utilized model for rogue waves [14,15]. This solution is only nonsingular in the focusing regime unless higher order terms are considered [16].
When there are two or more wave trains present, the governing model is then the system of coupled nonlinear Schrödinger equations [17,18]. The special case where all the coefficients of cubic (1) where δ describes the group velocity mismatch and σ measures the coefficient of cubic nonlinearity.
The objective of this paper is to investigate the dynamics of rogue waves and breathers in a multi-component system through a combination of theoretical perspective and computational approach. Theoretically, the formation of rogue wave and breathers can be explained in terms of modulation instability (MI). A detailed numerical investigation on the evolution of a plane wave perturbed by a single wavy disturbance would supplement the theoretical framework. In particular, a disturbance from the baseband of the MI spectrum would generate a rogue wave. For a mode with a finite frequency, both symmetric and asymmetric breathers are observed from the computational study.
The outline of the paper is as follows. In Section 2, the rogue wave mode is derived by the Hirota bilinear method. The coupling effect is discussed in Section 3. The generation of rogue wave mode from baseband modulation instability is confirmed directly by computer simulation in Section 4. Similarly, the generation of symmetric and asymmetric breathers is demonstrated in Section 5. In Section 6, a numerical method for finding rogue wave modes in non-integrable systems is proposed. The conclusion is drawn in Section 7.

Formulation of the Rogue Wave Modes
Under the transformations, (1) can be rewritten in terms of the Hirota bilinear operator [37] as where ω 1 = ω 2 = ω 3 = C, and C = σ ρ 2 1 + ρ 2 2 + ρ 2 3 . The methodology in deriving the rogue wave solution is similar to that used in our earlier works [31] and thus the details are omitted here. The amplitude of the plane wave background, , is taken to be identical for all three waveguides. Basically a breather is first derived using a two-soliton expression with complex conjugate wavenumbers. By taking the small frequency limit of the breather, the rogue wave mode is given by where a and b are the real part and imaginary part of Ω 0 : Ω 0 = a + ib. The parameter Ω 0 is the leading order term in the asymptotic expansion of the wavenumber, which satisfies the dispersion relation, This cubic polynomial in Ω 2 0 will dictate the dynamics and profiles of the rogue waves, to be highlighted in the following section.

Extension of Existence Regime
The rogue wave solution in Equation (2) is nonsingular if and only if a never vanishes.
Since Ω 0 cannot be real, this is equivalent to having a non-real root for the cubic polynomial By considering the discriminant of p 3 (x), complex roots (and hence rogue waves) will exist for 0 < δ 2 < 16.9 σρ 2 . One highlight of the present work is that this constraint is much less restrictive than the corresponding existence condition for the Manakov system with two components, namely, 0 < δ 2 < 4σρ 2 [31]. We conjecture that the addition of more components to coupled systems may in general induce further modulation instabilities and enhance the existence of rogue waves. This hypothesis obviously must be tested for other dynamical systems in the future.
Moreover, the incorporation of the third component increases the complexity of the geometry of the wave profiles. Analytically, the dispersion relation is expressed as a higher order algebraic polynomial and will allow multiple rogue wave solutions under the same input physical parameters. 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.

Enhancement of Amplitude
For the two-component Manakov system in the defocusing regime [31], the range of amplitude and configurations of the rogue waves in the two components are identical. Either both wave profiles are eye-shaped dark rogue waves (EDRW) with the main displacement below the background, or both patterns are four-petal-shaped rogue waves (FPRW) with two local maxima and two local minima. Moreover, the maximum and minimum values attained are identical in both components.
For the three components case, the scenarios are drastically different as the various components in Equation (1) can exhibit distinct forms of rogue waves and the maximum displacements can be different ( Figure 1). Interestingly, there must be at least one component in the form of an EDRW where the minimum amplitude is bounded below by about 0.4 (See Proof in Appendix A). However, the rogue wave solution cannot take the form of an eye-shaped bright rogue wave with the main displacement above the background (See Proof in Appendix A). As compared to the two-component system, the nonlinear coupling of the third component can enhance the rogue wave in two ways: increasing the maximum amplitude and 'squeezing' a bright rogue wave.
The maximum amplitude can be greater than √ 2ρ (Figure 1a), which is the upper bound of amplitude for the two-component Manakov system [31]. Similar increment in amplitude due to coupling was also found in other coupled systems such as the long wave-short wave resonance model with two short wave components [41] and a system of coupled derivative nonlinear Schrödinger equations [38].
Although the formation of eye-shaped bright rogue wave is also prohibited [31], a tendency towards the formation of bright rogue wave can be observed in the three-component system. Such bright type rogue wave has a four-petal configuration where the saddle point is above the background and is closer to the maximum than the minimum. The u-component in Figure 1a exhibits a bright four-petal configuration where the amplitude at the saddle point is about 1.3 and the rogue wave ranges from 0 to about 1.5. Such geometry closely resembles the widely studied eye-shaped bright rogue wave.

Baseband Modulation Instability
Baseband modulation instability, the instability due to low frequency disturbances, has been shown to be intimately related to the existence condition of rogue waves [29]. The connection was established theoretically in several dynamical systems [38,39,41,42]. This section focuses on the role of baseband modulation instability in the formation of rogue wave.

Analytical Approach
To study the correlation between rogue waves and modulation instability, plane waves with identical amplitude are considered, Focusing on the instability of low frequency disturbances ( This is identical to the dispersion relation Equation (3) with a slight change in variable:

Baseband Modulation Instability
Baseband modulation instability, the instability due to low frequency disturbances, has been shown to be intimately related to the existence condition of rogue waves [29]. The connection was established theoretically in several dynamical systems [38,39,41,42]. This section focuses on the role of baseband modulation instability in the formation of rogue wave.

Analytical Approach
To study the correlation between rogue waves and modulation instability, plane waves with identical amplitude are considered, Small perturbations of the form exp[i(Kt − Wz)] would be governed by Focusing on the instability of low frequency disturbances ( This is identical to the dispersion relation Equation (3) with a slight change in variable: confirming again the relation between baseband modulation instability and rogue waves.

Computational Approach
The generation of localized modes resembling rogue waves starting from a chaotic field initial condition was studied in the literature [14]. Such modes can only be generated in parameter regimes with baseband modulation instability [42]. Here we demonstrate the emergence of rogue waves from a plane wave perturbed by one single mode instead of a random noise. More precisely, we consider the initial condition where u 0 , v 0 and w 0 are the plane waves given in Equation (4). Equation (1) is numerically solved with a combination of pseudospectral method and a fourth-order Runge-Kutta scheme [43].
The result is illustrated with the typical case of σ = 10, = 1 and δ = 5. The modulation instability gain spectra exhibit multiple bands due to the existence of multiple complex roots of Equation (5) (Figure 2). For small frequency K, patterns resembling rogue waves are observed (Figure 3, only patterns for u are shown, as profiles for the other two waveguides are similar). If there exists a pair of rogue wave modes for the same input parameters, the rogue wave with a higher baseband growth rate would dominate the other mode [39]. 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].

Computational Approach
The generation of localized modes resembling rogue waves starting from a chaotic field initial condition was studied in the literature [14]. Such modes can only be generated in parameter regimes with baseband modulation instability [42]. Here we demonstrate the emergence of rogue waves from a plane wave perturbed by one single mode instead of a random noise. More precisely, we consider the initial condition where u0, v0 and w0 are the plane waves given in Equation (4). Equation (1) is numerically solved with a combination of pseudospectral method and a fourth-order Runge-Kutta scheme [43].
The result is illustrated with the typical case of σ = 10, ρ = 1 and δ = 5. The modulation instability gain spectra exhibit multiple bands due to the existence of multiple complex roots of Equation (5) (Figure 2). For small frequency K, patterns resembling rogue waves are observed (Figure 3, only patterns for u are shown, as profiles for the other two waveguides are similar). If there exists a pair of rogue wave modes for the same input parameters, the rogue wave with a higher baseband growth rate would dominate the other mode [39]. 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].

Computational Approach
The generation of localized modes resembling rogue waves starting from a chaotic field initial condition was studied in the literature [14]. Such modes can only be generated in parameter regimes with baseband modulation instability [42]. Here we demonstrate the emergence of rogue waves from a plane wave perturbed by one single mode instead of a random noise. More precisely, we consider the initial condition where u0, v0 and w0 are the plane waves given in Equation (4). Equation (1) is numerically solved with a combination of pseudospectral method and a fourth-order Runge-Kutta scheme [43].
The result is illustrated with the typical case of σ = 10, ρ = 1 and δ = 5. The modulation instability gain spectra exhibit multiple bands due to the existence of multiple complex roots of Equation (5) (Figure 2). For small frequency K, patterns resembling rogue waves are observed (Figure 3, only patterns for u are shown, as profiles for the other two waveguides are similar). If there exists a pair of rogue wave modes for the same input parameters, the rogue wave with a higher baseband growth rate would dominate the other mode [39]. 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].

Asymmetric Breathers
Breathers can be generated through these simulations by starting with a disturbance of higher frequency in the unstable band. For typical values of σ = 10, = 1 and δ = 5, there exist two pairs of complex conjugate roots for Equation (5) at the baseband and the instability growth rates are identical. Two breathers can be generated concurrently and superposition leads to an asymmetric breather ( Figure 4). For most rogue waves studied in the literature, the local extrema are usually symmetric with respect to the main displacement and attain the same value. For an asymmetric rogue wave or breather [27,44], symmetry is broken, e.g., the four-petal arrangement is destroyed and one of the minimum points splits into two minima (Figure 4a).

Asymmetric Breathers
Breathers can be generated through these simulations by starting with a disturbance of higher frequency in the unstable band. For typical values of σ = 10, ρ = 1 and δ = 5, there exist two pairs of complex conjugate roots for Equation (5) at the baseband and the instability growth rates are identical. Two breathers can be generated concurrently and superposition leads to an asymmetric breather (Figure 4). For most rogue waves studied in the literature, the local extrema are usually symmetric with respect to the main displacement and attain the same value. For an asymmetric rogue wave or breather [27,44], symmetry is broken, e.g., the four-petal arrangement is destroyed and one of the minimum points splits into two minima (Figure 4a).

Rogue Wave in Non-integrable Systems
In many physical applications, the coefficients for the self-phase modulation (SPM) and cross-phase modulation (XPM) are distinct from each other, analytical schemes will usually fail for such coupled nonlinear Schrödinger equations [17,18]. It will be instructive to apply the mechanism developed here for such 'non-integrable' equations. Based on the numerical solution, wave profile and amplification ratio of the rogue wave can be calculated.
We demonstrate the idea with a special case where the ratio of SPM to XPM is a constant: where γ is the ratio of XPM to SPM. Under the initial condition where K is a low frequency of the wavy disturbance, rogue wave modes are generated (Figures 5  and 6). Four-petal RW-like waveforms can be observed with γ = 2 ( Figure 5). As compared to Figure 3, the minima get closer and the saddle point attains a smaller intensity. Dark RW-like patterns are observed for the case where XPM is half of SPM ( Figure 6). Apparently, only one type of RW waveform can be observed.

Rogue Wave in Non-integrable Systems
In many physical applications, the coefficients for the self-phase modulation (SPM) and cross-phase modulation (XPM) are distinct from each other, analytical schemes will usually fail for such coupled nonlinear Schrödinger equations [17,18]. It will be instructive to apply the mechanism developed here for such 'non-integrable' equations. Based on the numerical solution, wave profile and amplification ratio of the rogue wave can be calculated.
We demonstrate the idea with a special case where the ratio of SPM to XPM is a constant: where γ is the ratio of XPM to SPM. Under the initial condition where K is a low frequency of the wavy disturbance, rogue wave modes are generated (Figures 5 and 6). Four-petal RW-like waveforms can be observed with γ = 2 ( Figure 5). As compared to Figure 3, the minima get closer and the saddle point attains a smaller intensity. Dark RW-like patterns are observed for the case where XPM is half of SPM ( Figure 6). Apparently, only one type of RW waveform can be observed. In practice, many evolution equations governing dynamical systems for laboratory and engineering settings are not integrable. Several approaches were demonstrated to be feasible as approximations or the estimations of properties of rogue waves in such non-integrable systems. For instance, analytical rogue wave solutions of the NLS equation with periodic modulated coefficients were taken as initial conditions in the numerical approximations of rogue waves for the non-integrable case with constant nonlinearity coefficient and periodic dispersion coefficient [45]. Furthermore, rogue waves from well-studied equations like the NLS equation can be utilized to study rogue waves in less thoroughly studied physical systems. With a suitable physical assumption, optical quadratic solution can be related to the solution of the NLS equation through the second-harmonic asymptotic expansion and the method of repeated substitution [46]. Hence, useful approximations of rogue waves in a quadratic medium can be obtained. It was shown that breathers and RW-like entities emerge from simulations with random initial conditions, which serve as an alternative methodology to examine rogue waves in a general nonlinear system.
In this work, an alternative method is proposed which is based on the generating mechanism of rogue waves. The method is independent of the exact rogue wave solution of the integrable system. Hence, this scheme should hopefully be quite widely applicable to general systems, and should not be restricted to the "quasi-integrable" regime where the coefficients are close to the "integrable" case. As compared to the detection of rogue waves in a chaotic field, the baseband In practice, many evolution equations governing dynamical systems for laboratory and engineering settings are not integrable. Several approaches were demonstrated to be feasible as approximations or the estimations of properties of rogue waves in such non-integrable systems. For instance, analytical rogue wave solutions of the NLS equation with periodic modulated coefficients were taken as initial conditions in the numerical approximations of rogue waves for the non-integrable case with constant nonlinearity coefficient and periodic dispersion coefficient [45]. Furthermore, rogue waves from well-studied equations like the NLS equation can be utilized to study rogue waves in less thoroughly studied physical systems. With a suitable physical assumption, optical quadratic solution can be related to the solution of the NLS equation through the second-harmonic asymptotic expansion and the method of repeated substitution [46]. Hence, useful approximations of rogue waves in a quadratic medium can be obtained. It was shown that breathers and RW-like entities emerge from simulations with random initial conditions, which serve as an alternative methodology to examine rogue waves in a general nonlinear system.
In this work, an alternative method is proposed which is based on the generating mechanism of rogue waves. The method is independent of the exact rogue wave solution of the integrable system. Hence, this scheme should hopefully be quite widely applicable to general systems, and should not be restricted to the "quasi-integrable" regime where the coefficients are close to the "integrable" case. As compared to the detection of rogue waves in a chaotic field, the baseband In practice, many evolution equations governing dynamical systems for laboratory and engineering settings are not integrable. Several approaches were demonstrated to be feasible as approximations or the estimations of properties of rogue waves in such non-integrable systems. For instance, analytical rogue wave solutions of the NLS equation with periodic modulated coefficients were taken as initial conditions in the numerical approximations of rogue waves for the non-integrable case with constant nonlinearity coefficient and periodic dispersion coefficient [45]. Furthermore, rogue waves from well-studied equations like the NLS equation can be utilized to study rogue waves in less thoroughly studied physical systems. With a suitable physical assumption, optical quadratic solution can be related to the solution of the NLS equation through the second-harmonic asymptotic expansion and the method of repeated substitution [46]. Hence, useful approximations of rogue waves in a quadratic medium can be obtained. It was shown that breathers and RW-like entities emerge from simulations with random initial conditions, which serve as an alternative methodology to examine rogue waves in a general nonlinear system.
In this work, an alternative method is proposed which is based on the generating mechanism of rogue waves. The method is independent of the exact rogue wave solution of the integrable system. Hence, this scheme should hopefully be quite widely applicable to general systems, and should not be restricted to the "quasi-integrable" regime where the coefficients are close to the "integrable" case. As compared to the detection of rogue waves in a chaotic field, the baseband disturbance can isolate the rogue-wave-like structures among entities such as breathers. A comprehensive study on rogue waves in a general multi-component system of coupled nonlinear Schrödinger equations will be carried out in the future. This numerical approach initiated from the insight gained from rational solutions should complement the limitation of analytical methods in the study of rogue waves.

Discussions and Conclusions
It was known that a multi-component Manakov system can effectively model wave propagation in a multicore optical fiber [22]. Besides temporal waveguides, such Manakov systems are relevant in other settings in optical physics too. For spatial solitons, diffraction will play the role of group velocity dispersion, and continuous variations of diffraction, nonlinearity, and gain/loss might lead to novel rogue wave patterns [47]. Similarly, Manakov soliton can arise for biased guest-host photorefractive polymer too [48]. Furthermore, reductive perturbation techniques can be employed to establish Manakov equations as approximations for propagation of electromagnetic fields along isotropic chiral metamaterials [49].
In this work, a theoretical study is performed to understand the increasing complexity of the Manakov models with larger number of components. It will be worthwhile if analytical and computational predictions here can be verified in experiments in the future. More precisely, rogue wave solutions for the defocusing three-component Manakov system with group velocity mismatch are derived by the Hirota bilinear method. The nonlinear coupling effect is highlighted, namely, the extension of existence regime of rogue waves and the enhancement of amplitude. The main focus of the work is to demonstrate the generation of rogue wave from baseband disturbance.
Recently, the onset of baseband modulation instability has been proven to be equivalent to the existence condition of rogue waves in several systems. In this work, the role of low frequency disturbance in the formation of rogue waves is further consolidated through numerical simulations. By perturbing the plane wave solution by a single mode disturbance with a small frequency, rogue wave modes with configuration similar to the analytical rogue wave solutions can be generated. This idea can be generalized to approximate rogue wave modes in non-integrable systems where most analytical methods fail. The proposed numerical methodology can greatly enrich our knowledge of rogue waves in such systems. Detailed investigations on evolution of rogue waves in general systems of coupled nonlinear Schrödinger equations without any restriction on coefficients would be conducted in the future. Moreover, modified NLS equation or the corresponding systems can be studied [50].
Secondly, the single mode wavy disturbance to a plane wave can generate breathers as well. It is well-known that the formation of breathers is closely related to MI. Depending on the MI spectrum, both symmetric and asymmetric breathers can be generated from a single mode wavy disturbance. For the case where there are multiple unstable bands, the unstable mode can generate multiple breathers which superimpose to form an asymmetric breather. On the other hand, a conventional symmetric breather is generated if there exists only one unstable band.
In conclusions, rogue wave and breather formation are closely related to the nature of the MI spectrum. Through the study of the MI spectrum supplemented with numerical simulations, more intriguing wave dynamics of general non-integrable systems can be revealed in the future. Theoretically, three-component or multi-component systems with variable coefficients and external potential can be further investigated through a similarity transformation [13]. Recently, nonlocal equations have been widely studied due to their PT symmetric property [51]. Many nonlocal evolution equations display remarkable similarities in comparison with intensively studied classical ones, e.g., the nonlocal NLS equation iA z + A tt + AA * (−t, z)A = 0, which can be analyzed by direct and inverse scattering techniques [51]. Indeed, the nonlocal NLS model (Equation (9)) also possesses an infinite number of conservation laws. Thus total self-induced potential A(t,z)A*(−t,z) over the entire spatial domain is conserved, but not the usual intensity AA*.
This feature may have implications on the size of the "elevation" and "depression" regions of the rogue waves [52]. Naturally the structure of the scattering problem and Painlevé property are slightly different from those of the classical NLS equation. These features and other extensions, e.g., coupled waveguides, higher order dispersion, and discrete models will likely constitute fruitful paths of research in the future.