Watch-hand-like optical rogue waves in three-wave interactions

Shihua Chen; Jose M. Soto-Crespo; Philippe Grelu

doi:10.1364/OE.23.000349

1. Introduction

In recent years rogue waves in optical systems have been the subject of an intense research since the seminal observation of optical rogue waves in a photonic crystal fiber [1]. They offer an excellent and convenient analogy to the oceanic rogue waves that are intrinsically difficult to monitor because of their fleeting existences [2–5]. Although some controversy still remains in defining extreme wave events, the optics community has adopted such general criteria that the rogue waves should have a high peak amplitude or a fast rise and fast fall signature and their occurrence should follow a skewed L-shaped distribution [1, 4, 5].

Nonlinear dynamics is one of the theoretical frameworks that has been successful in predicting the basic features of rogue waves [6, 7]. For example, the prototypical profile of the single rogue wave event in various experimental fields has frequently been provided by the so-called Peregrine soliton [2, 8], the simplest doubly-localized rational solution to the nonlinear Schrödinger (NLS) equation. Recently, the study of the rogue wave dynamics has been extended from the scalar model to coupled systems [9–13], and from the fundamental structures to the higher-order hierarchy [14–17], aiming at a deeper understanding of their manifestations. Particularly, there has been an emerging interest in studying the protean rogue wave phenomena in three-wave resonant interaction (TWRI) systems [10, 11].

TWRI is one of the fundamental parametric processes that occurs universally in any weakly dispersive medium whose lowest order nonlinearity is quadratic in terms of the wave amplitudes [18]. In optical contexts, it describes parametric amplification, frequency conversion, and stimulated Raman and Brillouin scattering, forming the basis of our understanding of many pattern-forming systems [19–22]. As early as 1970s, integrability of the governing equations was established and soliton solutions were identified (see [18] and references therein). These solitons are coherent localized structures that result from a dynamic balance between the energy exchanges due to the nonlinear interaction and the convection due to the group velocity mismatch [23], in contrast to the quadratic solitons for which the energy flow is counterbalanced by the group-velocity dispersion (or diffraction) [24]. Interestingly, the TWRI solitons can propagate in a common (or locked) velocity, despite the fact that the three field components travel with their respective group velocity before being mutually trapped [25, 26].

In this article, we investigate the novel dynamics of super rogue waves that can be triggered in such a TWRI system, based on the exact second-order rational solutions. We find new types of highly asymmetric optical rogue waves which always feature a watch trait, with each rogue wave hand (component) involving a peak amplitude more than five times the height of the surrounding background. We also provide numerical evidences for the robustness of these exotic rogue wave patterns in spite of the onset of modulational instability (MI).

2. The TWRI equations and exact second-order rogue wave solutions

The TWRI equation that governs the propagation of three optical pulses, perfectly phasematched in a weakly dispersive quadratic medium, can be written as [18–20, 25]

\begin{array}{l} u_{1 t} + V_{1} u_{1 x} = u_{2}^{*} u_{3}^{*}, \\ u_{2 t} + V_{2} u_{2 x} = - u_{1}^{*} u_{3}^{*}, \\ u_{3 t} + V_{3} u_{3 x} = u_{1}^{*} u_{2}^{*}, \end{array}

where u_n(x, t) (n = 1, 2, 3) are slowly varying complex envelopes of the three interacting optical fields, with t and x being the time and space variables, respectively. As in [11], the time t here is assumed to be the evolution variable. The subscripts x ant t stand for the partial derivatives and the asterisk denotes the complex conjugation operation. The coefficients V_n correspond to the relative group velocities of the three waves and we assume the ordering V₁ > V₂ > V₃. Without loss of generality, we set V₃ = 0, which implies writing Eqs. (1) in the reference frame comoving with u₃. The above choice of signs before quadratic terms is indicative of the nonexplosive character of the interaction [18]. We remark that Eqs. (1) are completely integrable and thus can be exactly solved by an array of standard analytical tools [10, 18, 23].

Considering the resonant conditions for the frequencies and momenta, Eqs. (1) allow the following coupled plane-wave solutions0

\begin{array}{l} u_{10} (x, t) = a_{1} \exp [- i (k_{1} x - ω_{1} t)], \\ u_{20} (x, t) = a_{2} \exp [i (k_{2} x - ω_{2} t)], \\ u_{30} (x, t) = i a_{3} \exp [i (k_{1} - k_{2}) x - i (ω_{1} - ω_{2}) t], \end{array}

where a₃ = a₁a₂/δ,

k_{1} = ω_{1} / V_{1} + a_{2}^{2} / (δ V_{1})

and

k_{2} = ω_{2} / V_{2} + a_{1}^{2} / (δ V_{2})

, with a_n being the respective background heights which are usually unequal. For convenience, we use A = (σ₁a₁)²+ (σ₂a₂)², B = (σ₁a₁)² − (σ₂a₂)²,

σ_{j} = \sqrt{V_{j} / (V_{1} - V_{2})} (j = 1, 2)

, κ = ω₁ + ω₂, and δ = ω₁ − ω₂. Such background fields are intrinsically unstable [27] and a tiny localized deformation can evolve into rogue waves or into other periodically-modulated wave structures.

The fundamental rogue wave solutions to Eqs. (1) were recently reported [10, 11]. For our present purposes, we are only concerned with the highly asymmetric super rogue waves formed at B = 0 and $δ = \sqrt{A / 2}$ , or equivalently, under the parameter condition

δ = σ_{1} a_{1} = σ_{2} a_{2} .

Using the nonrecursive Darboux transformation method (one can refer to [17] for technical details), the exact second-order rogue wave solutions can be found to be

\begin{array}{l} u_{1}^{[2]} = u_{10} {1 + \frac{3 i θ_{1} (λ_{0} - λ_{0}^{*}) [R_{0}^{*} (R_{1} m_{22} - S_{1} m_{21}) + S_{0}^{*} (S_{1} m_{11} - R_{1} m_{12})]}{σ_{1} a_{1} (m_{11} m_{22} - m_{12} m_{21})}}, \\ u_{2}^{[2]} = u_{20} {1 + \frac{3 i θ_{1} (λ_{0} - λ_{0}^{*}) [R_{2}^{*} (R_{0} m_{22} - S_{0} m_{21}) + S_{2}^{*} (S_{0} m_{11} - R_{0} m_{12})]}{σ_{2} a_{2} (m_{11} m_{22} - m_{12} m_{21})}}, \\ u_{3}^{[2]} = u_{30} {1 + \frac{3 i θ_{1} (λ_{0} - λ_{0}^{*}) [R_{1}^{*} (R_{2} m_{22} - S_{2} m_{21}) + S_{1}^{*} (S_{2} m_{11} - R_{2} m_{12})]}{σ_{1} σ_{2} a_{3} (m_{11} m_{22} - m_{12} m_{21})}}, \end{array}

where u_n₀ are the seeding plane-wave solutions (2),

λ_{0} = \frac{i \sqrt{6 A}}{4} + \frac{κ}{6}

,

θ_{1} = \frac{\sqrt{3}}{2} - \frac{i}{2} = θ_{2}^{*}

, and

\begin{array}{l} R_{0} = γ_{1} + 2 γ_{2} ξ + 4 γ_{3} (ξ^{2} + 2 ξ + 3 i ρ), \\ R_{j} = γ_{1} + 2 γ_{2} (ξ - \sqrt{3} θ_{j}) + 4 γ_{3} [ξ^{2} + 3 i ρ - 2 i {(- 1)}^{j} (ξ / θ_{j} - \sqrt{3})], \\ S_{0} = γ_{1} c_{0} + γ_{2} d_{0} + γ_{3} e_{0} + γ_{4} + 2 γ_{5} ξ + 4 γ_{6} (ξ^{2} + 2 ξ + 3 i ρ), \\ \begin{matrix} S_{j} = γ_{1} c_{j} + γ_{2} d_{j} + γ_{3} e_{j} + γ_{4} + 2 γ_{5} (ξ - \sqrt{3} θ_{j}) \\ + 4 γ_{6} [ξ^{2} + 3 i ρ - 2 i {(- 1)}^{j} (ξ / θ_{j} - \sqrt{3})], \end{matrix} \\ m_{11} = {| R_{0} |}^{2} + {| R_{1} |}^{2} + {| R_{2} |}^{2}, \\ m_{12} = R_{0}^{*} S_{0} + R_{1}^{*} S_{1} + R_{2}^{*} S_{2} - m_{11} \equiv m_{21}^{*}, \\ m_{22} = {| S_{0} |}^{2} + {| S_{1} |}^{2} + {| S_{2} |}^{2} - m_{12} - m_{21}, \end{array}

with γ_s (s = 1, 2, …, 6) being arbitrary complex constants. The other parameters are given by

\begin{array}{l} ξ = \frac{\sqrt{6 A}}{2} [t + i (\frac{1}{V_{1} θ_{1}} - \frac{1}{V_{2} θ_{2}}) x], ρ = \sqrt{2 A} (\frac{1}{V_{2}} - \frac{1}{V_{1}}) x, η = \sqrt{6 A} (\frac{1}{V_{2}} + \frac{1}{V_{1}}) x, \\ c_{0} = \frac{1}{6} ξ^{3} + ξ^{2} + \frac{1}{2} (η + 3 i ξ ρ), c_{j} = c_{0} - \frac{\sqrt{3}}{2} (ξ^{2} + 3 i ρ) θ_{j} - \frac{\sqrt{3} (ξ - 1)}{θ_{j}}, \\ d_{0} = \frac{1}{12} ξ^{4} + ξ^{3} + ξ^{2} + \frac{4}{3} ξ - \frac{9}{4} ρ^{2} + \frac{3 i}{2} ρ ξ (ξ + 2) + η ξ, \\ d_{j} = d_{0} - \frac{\sqrt{3}}{3} [ξ^{2} + 5 ξ + 9 i ρ + {(- 1)}^{j} i (4 θ_{j} - \sqrt{3} ξ)] [θ_{j} ξ - {(- 1)}^{j} i] - \sqrt{3} θ_{j} η, \\ e_{0} = 2 i (3 η + 2) ρ + 2 η ξ (ξ + 2) + \frac{1}{9} ξ {(ξ^{2} + 6 ξ + 9 i ρ)}^{2} - \frac{2}{45} ξ (ξ^{4} - 120 ξ - 60), \\ e_{j} = e_{0} - \sqrt{3} θ_{j} {(ξ^{2} + 3 i ρ)}^{2} - 4 \sqrt{3} η [θ_{j} ξ - {(- 1)}^{j} i] - \frac{12 \sqrt{3} i (ξ - 1) ρ}{θ_{j}} \\ + ξ (ξ^{3} - 6 ξ^{2} - 4) + {(- 1)}^{j} \frac{i \sqrt{3}}{3} (ξ - 2) (ξ^{3} + 12 ξ - 4) . \end{array}

We should point out that, in addition to the six structural parameters γ_s, the rogue wave patterns determined by Eqs. (4) only depend on the free parameters a₁ and on the relative group velocities V₁ and V₂. The value of κ does not affect the intensity distributions of the three interacting rogue waves. Without loss of generality, one can let the background height of u₁ be unity, i.e., a₁ = 1, and then, it follows that a₂ = σ₁/σ₂, a₃ = 1/σ₂, and δ = σ₁, which are uniquely defined by V₁ and V₂. It is also noteworthy that the variables ξ, ρ, and η in Eqs. (4)–(6) depend on the inverse of the group velocities only, and therefore, for a large enough V₁ (with V₂ fixed), the rogue wave patterns could not change any more. More intriguingly, by choosing appropriate sets of structural parameters, the analytical solutions (4) can exhibit highly asymmetric super rogue waves located around the origin, like the three hands of a watch. In the following, we mainly discuss these novel watch-hand-like (WHL) super rogue waves, although other complex patterns (such as rogue wave doublets, quartets, and sextets) exist as well.

3. Novel WHL rogue wave dynamics and discussions

3.1. The watch trait and energy change

For this end, we simply choose the structural parameters γ₃ = 1 (with all the others being set to zero), and let a₁ = 1, a₂ = 3, and $a_{3} = 2 \sqrt{2}$ , which implies that V₁ = 9 and V₂ = 1. Typical WHL rogue wave states are illustrated in Fig. 1, clearly showing that the three rogue wave components have a watch-hand-like distribution [see panels (d)–(f)]. The watch trait will be more obvious if these three components are superimposed together in one image [see panel (g) where for comparison we have normalized the wave fields to have the same background height]. This unusual distribution is markedly different from the super rogue wave distributions that have been found to date in other coupled nonlinear systems, where the waves always feature a major overlap of their spatiotemporal distributions (see for instance Figs. 3(a) and 3(b) in [17] related to the coupled rogue waves of the Manakov system). In this regard, as they do not significantly overlap in time and space, the three components of these super WHL rogue wave states could be efficiently separated using an appropriate filtering technique: this represents an important feature that would facilitate the experimental diagnostics and observation.

Fig. 1 WHL optical rogue waves formed at a₁ = 1, V₁ = 9 and V₂ = 1: (a)–(c) surface plots; (d)–(f) contour distributions. (g) shows the watch trait by superimposing the above three components in one image. Here γ₃ = 1 and the other structural parameters are zero.

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We understand that the main spatiotemporal orientations of the WHL rogue wave components are determined by their relative group velocities. For instance, V₃ = 0 implies that, in the reference frame of u₃, the hump orientation of the component u₃ is frozen at around x = 0 [see Fig. 1(f)]. Also, as one can verify, increasing the value of V₂ towards V₁ would change the orientation of the other two field components, but still would maintain a watch trait, apart from its hands being extended significantly along the x dimension. Nevertheless, the spatiotemporal profile of each rogue wave hand, although being so asymmetric and narrow, is still smooth from the mathematical point of view.

In addition to the above mentioned watch trait, all the three rogue wave components feature a giant wall-like hump, with a peak amplitude more than five times the respective background height. For illustration, we define an amplification factor g as the ratio of the peak amplitude to the average background. It is clearly seen in Figs. 1(a)–1(c) that, for the three rogue wave components, the corresponding g values are 5.43, 5.43 and 6.09 respectively. All of them are larger than 5, a characteristic value that the super (second-order) rogue waves in the scalar NLS equation can reach [6, 16]. Here we point out that, for other parameter scenarios where γ₃ = 0, the g value is found to be markedly smaller than 5, although usually a WHL distribution remains. Therefore, to form the super WHL rogue waves with g > 5, γ₃ should not be vanishing.

A further study suggests that the high-amplitude WHL super rogue waves arise indeed from the nonlinear superposition of six fundamental rogue waves, as seen in Fig. 2, where we used otherwise identical parameters as in Fig. 1, except for choosing another set of structural parameters as specified in the caption. Similar pertinent explanation was raised in [14] and is now being regarded as a suitable criterion for classifying the high-order rogue wave solutions [15]. Evidently, it is the peculiar interaction of these six fundamental, Peregrine-soliton-like, rogue waves that gives rise to the ultrahigh g values as exhibited in Figs. 1(a)–1(c). If γ₃ = 0, the number of the fundamental rogue waves involved would be reduced to four or two (which depends on which parameter, γ₂ or γ₁, is nonzero), and correspondingly the g values become smaller than 5. Basically, the less the number of fundamental constituents, the smaller the g value of the formed super rogue waves.

Fig. 2 Rogue wave sextets obtained under the same initial parameters as in Fig. 1, but with γ₁ = −200i, γ₃ = 1, γ₅ = 1000, γ₆ = 300, and γ₂ = γ₄ = 0.

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Fig. 3 Evolutions of the effective energy I_n (or P_n) with respect to t (or x) for: (a), (c) the WHL rogue wave case; (b), (d) the rogue wave sextet case.

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Let us simply comment on the energy exchanges among these three rogue wave components. We recall that Eqs. (1) represent an infinite-dimensional Hamiltonian dynamical system that would possess the following Manley-Rowe relations [11, 20, 25]:

\frac{d}{d t} (I_{1} + I_{2}) = \frac{d}{d t} (I_{2} + I_{3}) = 0,

where

I_{n} (t) = \frac{1}{2} \int_{- \infty}^{+ \infty} ({| u_{n} |}^{2} - {| u_{n 0} |}^{2}) d x (n = 1, 2, 3)

define the effective, renormalized, energy of the optical waves at a certain given time. We plot the energy evolutions in Fig. 3(a) for the super WHL rogue waves and in Figs. 3(b) for the rogue wave sextets, both sets corresponding to Figs. 1 and 2, respectively. We verify that these rogue waves, no matter how complex their structures are, always obey the energy relation I₁ = I₃ = −I₂. Indeed, important energy exchanges have occurred among these three wave components, in marked contrast to the case of coupled NLS equations where energy exchanges are excluded [28]. On the other side, if we define the effective pulse energy as

P_{n} (x) = \frac{1}{2} \int_{- \infty}^{+ \infty} ({| u_{n} |}^{2} - {| u_{n 0} |}^{2}) d t

, then, owing to symmetry, there are similar energy relations:

\frac{d}{d x} (V_{1} P_{1} + V_{2} P_{2}) = \frac{d}{d x} (V_{2} P_{2} + V_{3} P_{3}) = 0.

It is easily concluded from Eq. (8), with V₃ = 0 and the boundary conditions being taken into account, that the former two rogue wave components would have a zero pulse energy, i.e., P₁ = P₂ = 0, while the third one, P₃, has a nonzero evolution of energy with distance, as seen in Figs. 3(c) and 3(d), where the above two rogue wave types are considered correspondingly. As seen, both energy representations suggest an even stronger burst of energy in the super WHL rogue waves than in the sextets. However, despite the different evolution properties, the total energy in space and time should be conserved and equal to zero, i.e., $\int_{- \infty}^{+ \infty} I_{n} d t = \int_{- \infty}^{+ \infty} P_{n} d x = 0$ , which is a natural outcome of the Hamiltonian TWRI system (1).

3.2. Numerical simulations and modulational instability

Finally, for practical purposes, we numerically study the evolution dynamics of super WHL rogue waves as well as their stability against small perturbations, based on the standard split-step Fourier method [12, 13, 17]. Figures 4(a)–4(c) show the numerical results of the unperturbed WHL rogue waves, using identical parameters as in Fig. 1 and with the analytical solutions (4) at t = −10 as initial conditions. It is clearly seen that our numerical solutions are exactly consistent with the analytical solutions shown in Figs. 1(d)–1(f), at least within the propagation of 20 time units. This no doubt manifests the accuracy of our numerical scheme and its wide applicability to the present TWRI equation model.

We then perturbed the initial profiles by small amounts of white noise in order to test the stability of these super rogue waves. Specifically, we multiplied the real and imaginary parts of all the three field components u_n at t = −10 by a factor [1 + εr_i(x)] (i = 1, …, 6), respectively, where r_i are six uncorrelated random functions uniformly distributed in the interval [−1, 1] and ε is a small parameter defining the noise level. Shown in Figs. 4(d)–4(f) are the numerical results, for which we used a noise level of ε = 10⁻⁸. As seen, with this tiny perturbation, the above WHL rogue waves can propagate very neatly for a rather long time till the MI of the background fields grows up. We also simulated the rogue wave evolutions with a larger noise level or by initiating the profiles at larger negative values of time. It was shown that, despite that the field component u₂ will be overwhelmed by the periodic waves induced by the MI, the components u₁ and u₃ still manifest themselves clearly. This is not surprising because the MI, as revealed below, can grow exponentially with the propagation time and eventually forms large periodic wave structures that tend to strongly interfere with the trailing edge of the second rogue wave component, as claimed in [12, 13, 17].

Fig. 4 Simulations of the WHL optical rogue waves shown in Fig. 1 under otherwise identical parameter conditions. (a)–(c) show the unperturbed numerical results, while (d)–(f) are results obtained by perturbing the initial profiles with white noise.

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As a matter of fact, under the parameter condition (3), the MI of the background fields can be exactly evaluated. As in [12], we add small-amplitude Fourier modes to the plane-wave solutions (2) and express them as $u_{n} = u_{n 0} {1 + p_{n} \exp [- i Ω (μ t - x)] + q_{n}^{*} \exp [i Ω (μ * t - x)]} (n = 1, 2, 3)$ , where p_n and q_n are small amplitudes of the Fourier modes, and the parameters Ω and ε are assumed to be positive and complex, respectively. A substitution of these perturbed plane-wave solutions into Eqs. (1) followed by linearization yields the dispersion relations

Ω μ (μ - V_{1}) (μ - V_{2}) \pm \frac{\sqrt{2 A}}{2} [(\frac{V_{1}}{V_{2}} - 1 + \frac{V_{2}}{V_{1}}) μ^{2} - (V_{1} + V_{2}) μ + V_{1} V_{2}] = 0,

which consist of two cubic equations of ε. These cubic equations can be exactly solved using the Cardano’s formulas and hence the growth rate of the MI, defined by γ = ΩIm(ε), can be readily calculated. Figure 5(a) shows the combined map of the MI gains in the plane (V₂, Ω), calculated from Eq. (9) with given parameters a₁ = 1 and V₁ = 9. It is clear that the MI can always occur for 0 < V₂ < V₁ and the modulation frequency Ω_max at which the growth rate becomes maximum would be decreasing as V₂ → V₁, of course excluding the vicinity of V₁ [see panel (a)]. For our selected parameters (V₁ = 9, V₂ = 1), the modulation frequency Ω_max is around 1.1 [see panel (b)]. On the other side, we notice from our simulations shown in Figs. 4(d)–4(f) that the period of the MI-induced waves is around 24/4.5, corresponding to a modulation frequency of 3π/8 ≃ 1.18, almost the same as analytically predicted. This good consistency confirms further the soundness of our numerical results.

Figure 5(a) also shows that as V₂ grows towards V₁, the WHL rogue waves can be immune to stronger noise perturbations than those used in Figs. 4(d)–4(f), because in this case the growth rate of the MI is significantly reduced along with a low modulation frequency. We illustrate in Fig. 6 the simulated results of these super rogue waves at V₂ = 4, initially perturbed by white noise of intensity ε = 5×10⁻⁴. The other parameters are kept the same as in Figs. 4(d)–4(f). It is seen that each rogue wave hand can manifest itself very clearly within such a strong noise level, conforming to our analytical predictions above. Of course the modulation frequency indicated in Fig. 6, which is around 2π/15, is also well consistent with that shown in Fig. 5(a). These simulations, together with those shown in Fig. 4, suggest that our WHL rogue waves themselves are stable enough to develop for a wide range of parameters, although the background fields where they are built are always unstable due to MI.

Fig. 5 (a) The map of the MI gain versus V₂ and Ω for a₁ = 1 and V₁ = 9. The superimposed image in (a) results from calculations of two cubic equations in Eq. (9). (b) shows the growth rate (green line) versus Ω for a specific V₂ = 1, as indicated by the dash-dotted line in (a). The blue cross in (b) indicates the maximum value of the growth rate.

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Fig. 6 Simulations of the WHL optical rogue waves formed at a₁ = 1, V₁ = 9, and V₂ = 4, initially perturbed by white noise of intensity ε = 5 × 10⁻⁴.

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4. Conclusions

In conclusion, we presented the exact second-order rogue wave solutions to the TWRI equations which govern the resonant interaction of three optical pulses in quadratic nonlinear media. Among these analytical solutions, peculiar WHL super rogue wave patterns were demonstrated. All three waves of the coupled system involve a peak amplitude more than five times their respective background height, as well as a giant wall-like hump. The important spatiotemporal separation of the three giant wall humps represents a distinct feature of these coupled super rogue waves. We attributed such peculiar rogue wave structures to the nonlinear superposition of six fundamental (Peregrine-type) rational solitons, mediated by a particular process of energy exchange. It was revealed that they can exist for a wide range of parameters, as long as γ₃ ≠ 0 and other structural parameters are moderate with respect to γ₃. We confirmed by numerical simulations that these WHL rogue waves are stable enough to develop in spite of the onset of MI of the background fields. An analysis of the latter issue (i.e., MI) showed further that such exotic rogue wave patterns can indeed be immune to stronger noise perturbations as V₂ grows towards V₁. We anticipate that this relative stability along with the non-overlapping distribution property may facilitate the experimental diagnostics and observation of these super rogue waves in quadratic crystals [20, 26] or in laser-plasma interactions [22].

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grants No. 11174050 and No. 11474051). Ph. G was supported by the Agence Nationale de la Recherche (projects ANR-2010-BLANC-0417-01 and ANR-2012-BS04-0011). The work of J. M. S. C. was supported by MINECO under contracts FIS2009-09895 and TEC2012-37958-C02-02, and by C.A.M. under contract S2013/ MIT-2790.

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Watch-hand-like optical rogue waves in three-wave interactions

Abstract

1. Introduction

2. The TWRI equations and exact second-order rogue wave solutions

3. Novel WHL rogue wave dynamics and discussions

3.1. The watch trait and energy change

3.2. Numerical simulations and modulational instability

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (6)

Equations (9)

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