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along the y axis 

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where y = y$_{d}$/y$_{s}$ is the normalized coordinate, and we assume the same scaling of the transverse spatial dimen- sions, y$_{s}$ = x$_{s}$ . Then, the coupling phase coefficients are defined similar to Eq. (3), 

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where x$_{0}$ ( z ) and y$_{0}$ ( z ) define the transverse bending pro- file. 

We further consider the helical waveguide bending pro- file [see Fig. 4(a)], 

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where x$_{0}$$_{m}$ = y$_{0}$$_{m}$ is the bending amplitude (normalized to x$_{s}$ ) and L is the normalized bending period. 

Following the approach described above in Sec. II B, we consider wavepackets that are slowly varying in space and describe their profiles with a continuous function, a$_{n,m}$ ( t, z ) = ˜ u ( x = nd, y = md, t, z ). Then, we apply the scaling coefficients according to Eq. (9), and derive the approximate continuous equation [cf. Eq. (10)]: 

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where λ$_{x}$ ( z ) and λ$_{y}$ ( z ) the normalized diffraction coeffi- cients corresponding to the two transverse directions: 

Next, we perform the variational analysis using the three dimensional Gaussian Ansatz in the following form: 

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Following similar procedure as we used in the two- dimensional case [Sec. II C], we arrive at the set of vari- ational equations 

where E ≡ A $^{2}$W V T . 

We perform numerical analysis to identify periodic so- lutions of the variational equations (30). Such solutions correspond to three-dimensional light bullets, where the light-bullet extension along all dimensions ( x, y, t ) is re- stored after each bending period. We present an example of such solution in Fig. 4(b), which illustrates a stable and mobile three-dimensional light bullet. Indeed, the light bullet spatial and temporal localization widths are re- stored to the same values after each bending period.The solid line at the bottom shows the transverse position x$_{2}$ of the pulse center as a function of the propagation distance z for a finite initial beam tilt. The light bullet wobbles in the transverse direction during the propaga- tion, and it is clear that on average the pulse moves across the array. 

We summarize the existence and stability properties of three-dimensional light bullets in Figs. 4(c,d). We show the dependence between the energy E and the diffrac- tion map strength S on the phase increment Δ φ for one bending period. In comparison with the two-dimensional case [Fig. 1(d)], this dependence is much more complex, forming a loop-like structure. Interestingly, stable so- lutions, depicted with a solid line, are present for both positive and negative curve slopes of the E (Δ φ ) depen- dence, which indicates that the Vakhitov-Kolokolov cri- terion [36] is not applicable here. 

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