Abstract
The existence of stable solitons in two-and three-dimensional (2D and Typeset using REVT E X 1

of the Kerr coefficient, we consider the case when only the refractive index is modulated, while the nonlinearity is uniform. Soliton dynamics in a 1D version of this optical system, i.e., as a matter of fact, a multichannel nonlinear planar waveguide, was studied earlier in Ref. [17] (after the submission of the original version of this paper, a preprint has appeared which considers a very similar PhC model for the 2D case [18] ).

The present model is based on the rescaled Gross-Pitaevskii (GP) equation [19] for the wave function u(r, t),

(in optical models, it is the NLS equation, t being the propagation distance). Here, V (r) is the external potential, which contains the parabolic-trap and OL terms, V (r) = ω 2 /2 r 2 + ε[cos(2x) + cos(2y) + cos(2z)], r ≡ {x, y, z} ,

i.e., the OL period is normalized to be π (in the optical model, the parabolic trap is absent).

The only dynamical invariant of Eq. (1) is the norm (number of atoms in BEC, or the beam's power in the optical model), N = |u(r)| 2 dr.

Analytical predictions for solitons can be obtained by means of the variational approximation (VA), which was developed in nonlinear optics (see a review [20] ), and successfully applied to BECs too [3, 6, 21] , including BECs trapped in OL [22] (VA in the 1D version of this model was earlier elaborated for the above-mentioned multichannel nonlinear optical waveguide [17] ). Following these works, we adopt the Gaussian ansatz for fundamental solitons,

with an arbitrary negative constant µ and positive ones a and A (we set ε > 0 in Eq. (2), hence the central point of the ansatz must be chosen at the potential minimum, r = 0). In the case of BEC, µ is the chemical potential, and in optics −µ is the propagation constant.

Substituting the ansatz into the Lagrangian L corresponding to Eq. (1), we perform spatial integration, and derive equations for a and A in the form ∂L/∂a = ∂L/∂A = 0 (µ is not a variational degree of freedom, but rather an intrinsic parameter of the soliton family).

Below, we display variational results for ω = 0 in the GP/NLS equation (1) , which implies that the soliton has a size much smaller than that imposed by the parabolic trap.

In the 2D case, the variational procedure amounts to an equation which determines a in terms of the norm, N = πA 2 /a [this expression corresponds to the ansatz (3)], and another equation that yields the chemical potential/propagation constant:

Note that, if ε = 0, Eqs. (4) yield N = 4π. In the present notation, it is nothing else but a known variational prediction for the critical norm in the 2D NLS equation, which simultaneously is a universal value of the norm corresponding to the unstable 2D soliton.

With ε = 0, Eq. (4) shows that the number of particles attains a minimum (threshold) value,

at a = 1/2. This means that there is a threshold value of the norm which is necessary to create the 2D soliton; however, the threshold really exists only if it is positive. The expression (5) shows that the threshold exists for a relatively weak lattice, and it disappears if ε exceeds the value ε 0 = e 2 /8 ≈ 0. 92. 

Both the numerical and analytical solutions demonstrate that one branch [µ 1 (N) in Eq. (6) A basic conclusion following from the simulations is that the initial waveforms, taken 6 as predicted by VA, always evolve into stable solitons (in both 2D and 3D cases). On the other hand, any input with the norm below the threshold predicted by VA (if ε < ε 0 ), and some other input forms, taken very far from the variational prediction, decay into linear waves. This general result is illustrated by Fig. 1(b) from which we see that a delocalizing transition, manifested by the rapid drop of the amplitude, occurs at N cr /4π ≈ 0.62 for the case ε = 0.4. By comparing this value with the minimum value attained by N/4π along the VA existence curve of Fig. 1(a) at ε = 0.4 (i.e. N min /4π ≈ 0.58), we see that, although not perfect, the agreement is reasonable. Similar results are found also for other values of the amplitude of the optical lattice. We remark, however, that non-sharpness of the transition (see Fig. 1(b) ) makes it difficult to determine the threshold with higher accuracy since, close to the threshold, long simulations are necessary to check whether a broad small-amplitude soliton is formed or not. Accurate study of the threshold norm versus ε is presently under investigation and will be reported elsewhere.

The presence of a lower threshold for the existence of the multi-dimensional solitons in the OL strongly resemble a similar phenomenon occurring in nonlinear lattices for intrinsic localized modes [23] . This analogy correlates with the fact that the above solitons can be strongly localized around one site of the lattice (see below).

Actually, the lattice stabilizes the soliton not only against decay, but also partly against collapse. Indeed, in Fig. 1(b) the norm extends to slightly overcritical values (the exact value in 2D is N cr /4π = 0.93 [15] ) without provoking the collapse. An explanation for this comes from the fact that the chemical potential of the soliton lies in the gap of the excitation spectrum created by the lattice, thus enhancing their stability against the collapse or decay into linear waves. The mechanism for the formation of gap solitons in BECs with the OL and positive scattering length was identified as modulational instability of the Bloch states at the edges of the Brillouin zone [11, 12] .

Depending on the strength of the lattice potential and norm of the initial pulse, the emerging soliton occupies a single lattice cell (then we call it a "single-cell soliton"), or spreads itself over a few cells (a "multi-cell soliton"). Single-cell solitons always have a larger norm than the multi-cell ones, and a border value of N, which separates the singleand multi-cell states, can be approximately identified. The relaxation of the 2D and 3D pulses into the stable shape proceeds with oscillations [see inset to Fig. 1(a) ], with a higher frequency in stronger lattices. In fact, the input which is predicted by VA to be a stable soliton (according to the VK criterion) quickly evolves into a single-cell soliton whose shape is quite close to the predicted one, which is illustrated by inset to Fig. 1a . In contrast to this, the initial pulse taken as what is expected to be a VK-unstable soliton, according to VA, undergoes violent evolution, with large emission loss, which ends up by the formation of a multi-cell soliton. Full evolution pictures are not displayed here, as it is difficult to generate them in the 2D and 3D geometry without radial symmetry.

Generic examples of the established single-cell and multi-cell 2D solitons, which were generated, respectively, by the VA-predicted VK-stable Gaussian ansatz in a moderately weak lattice potential, and the VK-unstable ansatz in a strong potential, are displayed in The formation of solitons in the 3D lattice potential is, generally, similar to the 2D case, 8 although the relaxation time is shorter, due to the stronger interaction involving a larger number of adjacent cells. A typical example of a stable multi-cell 3D soliton, which forms itself in a strong lattice potential, is given in Fig. 3(a) .

Alongside the fundamental 2D soliton, in the absence of the lattice one can construct vortex solitons, of the form u = v(r) exp (−iµt + iSθ), where r and θ are the polar coordinates, S is an integer vorticity ("spin"), and v(r) is a real function, which exponentially decays as r → ∞ and vanishes ∼ r S as r → 0. As well as the fundamental (S = 0) soliton, the vortices are strongly unstable in the usual 2D NLS equation. Recently, stable vortices have been found in 2D (see reviews [24] ) and 3D [25] models with non-Kerr optical nonlinearities. On the other hand, it was demonstrated [27] that stable vortices, solely with S = 1, exist in the discrete version of the usual cubic 2D NLS equation (which describes a bundle of nonlinear optical waveguides [26] , or BEC trapped in a strong OL field [22] . Note that, unlike the isotropic NLS model, in ones with the axial symmetry broken by the lattice the vorticity is not a topological invariant, hence the very existence of such solutions is a nontrivial issue [27] .

To generate vortex solitons, we simulated the 2D version of Eq. (1) with the initial condition u 0 (r, θ) = Ar S exp(−ar 2 ) exp(iSθ), trying various values of A and a. Stable vortices with S ≥ 2 could never be created (they quickly spread out); however, initial configurations with S = 1 readily self-trap into a stable vortex, which features a complex multi-cell organization, see Fig. 3 (b). In conclusion, we have demonstrated that the GP/NLS equation with the attractive cubic nonlinearity and periodic potential, which describes BECs trapped in a 2D or 3D optical lattice, and (in the 2D case) an optical beam in the Kerr medium with a transverse periodic modulation of the refractive index, gives rise to stable solitons. In moderately weak and strong lattices, single-cell and multi-cell solitons were found, respectively, the former ones being accurately predicted by the variational approximation. A necessary condition for the formation of 2D solitons in a relatively weak lattice is that the initial norm of the field must exceed a threshold value. Stable 2D vortex solitons with S = 1 were found too.

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