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Spatiotemporal discrete multicolor solitons

We have found various families of two-dimensional spatiotemporal solitons in quadratically nonlinear waveguide arrays. The families of unstaggered odd, even and twisted stationary solutions are thoroughly characterized and their stability against perturbations is investigated. We show that the twisted and even solutions display instability, while most of the odd solitons show remarkable stability upon evolution.


I. INTRODUCTION
Since their first experimental observation [1], quadratic solitons have been demonstrated in a variety of materials and geometries. Spatial, temporal, and spatiotemporal solitons in quadratic media have been extensively investigated both experimentally and theoretically (for detailed reviews, see [2,3,4,5]). Quadratic solitons also exist in the form of discrete entities, namely strongly localized wave packets forming in nonlinear waveguide arrays. Since their theoretical prediction in 1988 in cubic nonlinear media [6], discrete optical solitons have attracted a steadily growing interest because of their potential applications in switching and routing devices [7,8,9]. The discrete solitons that form in tight-coupled waveguide arrays made of quadratic nonlinear media have been comprehensively investigated [10,11,12,13,14,15] due to the rich variety of effects that are possible with them. Such richness may be further enhanced by combining the features of both, continuous and discrete soliton families present in spatiotemporal discrete solitons, a possibility that we address here.
In the last two decades the concept of optical spatiotemporal solitons (STSs), referred as light bullets in the three-dimensional case [16], has been attracting attention as a unique opportunity to create a self-supporting fully localized object. The existence of STS's in quadratic nonlinear materials was theoretically predicted [17] and thereafter experimentally realized in a two-dimensional geometry involving one temporal and one spatial coordinate [18]. The existence and properties of continuous-discrete spatiotemporal solitons has been extensively investigated in cubic nonlinear media and stable odd solitons have been shown to exist [19,20,21,22,23]. It was shown that the cubic weakly-coupled waveguide arrays act as collapse compressors [19,20,21]. In contrast with the cubic spatiotemporal solitons, the quadratic ones do not display collapse in both two-and three-dimensional geometries [24].
A still open problem, not analyzed so far, is the existence of space-time solitons in nonlinear waveguides with quadratic nonlinearity, that is, the existence of discrete spatiotemporal multicolor solitons.
In this paper we investigate in detail the existence and stability of three representative families of two-dimensional spatiotemporal solitons in quadratic nonlinear waveguide arrays.
We assume, in addition to the temporal dispersion of the pulse, the contribution of the discrete diffraction, that arises because of the weak coupling between neighboring waveguides.
Discrete soliton solutions were classified as staggered and unstaggered ones (see, for ex-ample, Ref. [25]). The staggered solutions display out-of-phase fields between the neighbor noncentral waveguides whereas the unstaggered ones display in-phase fields in these noncentral waveguides. Inside each of these classes of solitons (staggered and unstaggered) one can find solutions with different topologies, dictated mainly by the energy and phase distribution in the central waveguides. Thus, one can have (i): odd solitons, for which most part of the energy is located in one central waveguide and the energy distribution across the waveguide array is symmetric with respect to this central waveguide, (ii): even solitons, for which most part of the energy is equally distributed in the two central waveguides, the fields in these central waveguides being in-phase and of equal amplitudes, and (iii): twisted solitons, for which most part of the energy is equally distributed in the two central waveguides, but the fields in the two central waveguides are out-of-phase.
Here we will restrict ourselves to three representative families of continuous-discrete unstaggered solitons, namely the odd soliton (see Fig. 1(a)), the even soliton (see Fig. 1(c)) and the twisted soliton (see Fig. 1(d)). Note that for the twisted soliton, the fundamental frequency field is, in fact, an anti-symmetric one (the π jump of phase occurs only between the two central waveguides), whereas the second harmonic field is a symmetric one (having the form of an even discrete soliton). For all the solutions we deal with, the temporal profile, i.e. the shape of the pulses propagating in a specific waveguide, is a bell-shaped symmetric one (see Fig. 1(b), below). Besides these stationary solutions, there exist a whole "zoology" of localized solutions, including staggered solitons, dark or dark-bright solitons, but their study is beyond the scope of the present work.

II. MODEL AND STATIONARY SOLUTIONS
The evolution of the spatiotemporal two-component field in quadratic nonlinear waveguide arrays in a degenerate second-harmonic generation geometry may be described by the following set of nonlinearly coupled reduced differential equations: where u n and v n represent the normalized amplitudes of the fundamental frequency (FF) and second-harmonic (SH) fields in the nth waveguide, with n = −N, ... − 1, 0, 1, ..., N, 2N + 1 being the number of waveguides, * means complex conjugation, c u,v and g 1,2 are the linear coupling coefficients and group-velocity dispersion (GVD) coefficients, respectively, and β is the wave-vector mismatch. The evolution variable ξ denotes the normalized propagation distance along the waveguides. The dynamical system (1) admits several conserved quantities including the energy flow and Hamiltonian which read where we have defined A n ≡ u n , and B n ≡ v n exp (−iβξ). The stationary solutions of V n (τ ) are real functions, and b 1,2 are real propagation constants verifying b 2 = 2b 1 + β.
Continuous-discrete solitons arise from a balance between discrete diffraction, dispersion and quadratic nonlinearity. The families of odd, even, and twisted stationary continuousdiscrete solitons have been obtained numerically by a standard relaxation method. For given coupling strengths c u,v , dispersions g 1,2 and wave-vector mismatch β, the solitons families are parametrized by the nonlinear wavenumber shift b 1 . The coupling coefficients c u,v were considered positive, and equal, so further we introduce single parameter C to describe coupling between neighboring guiding sites. Throughout this paper we will always consider anomalous dispersions at both frequencies and we fixed g 1 = −0.25 and g 2 = −0.5. Note that in the continuous case, long-lived soliton-like propagation when the GVD is slightly normal at SH is known to occur, [26,27] thus a similar behavior might occur in the continuousdiscrete spatiotemporal case analyzed here.
In Fig. 2(a,b) we show the dependencies of the peak amplitude A u and the temporal full narrower than those forming for smaller ones. This feature was observed for one-and two-dimensional continuous solitons in quadratic media for which at phase matching the product peak-amplitude × width-squared is a constant quantity [28]. In Fig. 2(d) we plot the amplitude of the stationary odd soliton as function of its temporal width. We see that outside phase-matching the families of solitons exhibit a more complicated amplitude-width relationship, similar to the case of continuous quadratic solitons [28]. The scaling properties of Eqs. (1) can be written as: where ψ being the scaling parameter. to be the most robust on propagation. The Peierls-Nabarro potential, that is the difference between Hamiltonian of the odd soliton and that of the even one [29], corresponding to the same energy flow, is negative everywhere. From a geometrical point of view, this would mean that odd solitons are stable in the entire domain of their existence [30]. Our numerical simulations, described in detail in the next section, show that, indeed, this is the case except for solitons at negative phase-mismatches that are unstable only in a narrow region near cut-off (see Fig. 5(a)) [28,31].

III. STABILITY ANALYSIS
A key issue concerning the soliton families we found is their stability on propagation.
In order to elucidate if the localized continuous-discrete solitons are dynamically stable we have performed both a linear stability analysis and direct numerical simulations. We seek for perturbed solution of Eq. (1) in the form Here µ is a small parameter, U n (τ ) and V n (τ ) are the stationary solutions and f n (τ, ξ) and h n (τ, ξ) are the perturbations. Then after linearizing the evolution equations (1) we are left with a system of linear coupled differential equations for the perturbations (see, e.g., Ref. [32]): We have solved both this linear system and the nonlinear dynamical equations (1) with a combined Fast-Fourier Transform, to deal with the linear differential part in the temporal coordinate, and a fourth-order Runge-Kutta method, to deal with the cross-coupling terms.
We have typically used 512 or 1024 points in the time domain and we have considered tens of array sites (e.g., 61), depending on the width of the solution whose stability is investigated. In order to let the radiation to escape from the computation window we have implemented transparent (absorbing) boundary conditions. We multiply a flat-top function after every step of longitudinal propagation distance, and this function has a very narrow tail which is zero.
We have determined the dominant eigenvalue δ of the linearized problem using the same approach as in Ref. [32]. The method gives us only the dominant eigenvalue, not the whole eigenvalue spectrum. This eigenvalue corresponds to the most rapidly (exponentially) developing instability. The noisy perturbation we consider at ξ = 0 develops, during evolution, to a localized eigenvector with a well defined symmetry, depending on the type of the solution considered. In the cases where an instability was detected, only real instability eigenvalues were found. The dominant eigenvalue was calculated in the form This dominant eigenvalue tends to zero when one approaches the stability region. The Our calculations show that odd continuous-discrete solitons obey the Vakhitov-Kolokolov stability criterion [33], i.e., they are stable provided dI/db 1 > 0, and unstable, otherwise.
The Vakhitov-Kolokolov criterion was shown also to hold for discrete space-time solitons that exist in Kerr nonlinear media [22,23]. Moreover, the unstable odd cubic continuous-discrete solitons can display collapse-type instabilities, a reminiscent feature of the two-dimensional stationary solutions of nonlinear Schrödinger equation, while the unstable quadratic discrete space-time odd solitons do not display this type of instability [24].
Let us stress that as compared to the one-dimensional discrete twisted solitons forming in quadratic media that can be stable, in specific parameter regions, in our case, the introduction of a time coordinate leads to the destabilization of these solutions. However, one of the central points of this work is that we found families of stable odd continuous-discrete multicolor solitons. As illustrated in Fig.6 (b), stable odd solitons can propagate for huge distances without altering their shape and eliminating the added random white noise during evolution. The case shown here corresponds to negative wave-vector mismatch β = −3 but similar stable evolution has been obtained for positive mismatches and phase-matching geometries except for odd solitons from the branch, where dI/db 1 < 0, which are unstable and will therefore decay after a finite propagation distance (see Fig. 6(a)).
In addition, we also thoroughly investigated the decay scenarios of the other two types of solitons: even and twisted. As stated before we have not observed any stable even or twisted continuous-discrete soliton. Fig. 7 shows possible instability scenarios for unstaggered even and unstaggered twisted solitons. We have found that perturbed even soliton typically tranforms into odd one through increasing field oscillation in neighboring wave guides ( Fig. 7(a)), and perturbed twisted soliton usually splits into two solitons which fly apart as when a repulsive force would act between them ( Fig. 7(b)). We have observed a π phase difference between the formed odd solitons and this could explain the repulsive force between them. Note that during the splitting process the resulting odd solitons are still locked in a specific waveguide, they are being allowed to repel in time. This unique feature comes with discreteness which does not allow the soliton energy to escape from the waveguide where it was initially located.

IV. CONCLUSION
We have shown that stable, spatiotemporal continuous-discrete solitons are possible in quadratic nonlinear waveguide arrays. Families of unstaggered odd, even and twisted sta-