2 and finally obtain We further consider the harmonic waveguide bending profile, in Eq. (1) to be dimensionless, applying the following normalization: where A$_{n}$ and variables with the subscript d denote the values in physical dimensions, and subscript s denotes the scaling coefficients. With these notations, the coupling phase coefficient is defined as where x$_{0}$$_{m}$ is the bending amplitude (normalized to x$_{s}$ ) and L = L$_{d}$/z$_{s}$ is the normalized bending period. To reduce the number of independent coefficients in Eq. (6), we choose the scaling coefficients as follows: where x$_{0}$ ( z ) is the waveguide bending profile normalized to x$_{s}$ , prime stands for the derivative, d$_{x}$ = d$_{d}$/x$_{s}$ is the normalized transverse separation between the neighbor- ing waveguides, λ$_{d}$ is the laser wavelength in vacuum, and n$_{0}$ is the average medium refractive index. We note that the values of the scaling coefficients x$_{s}$ , z$_{s}$ , t$_{s}$ , and I$_{s}$ can be chosen arbitrarily. Accordingly, the results ob- tained for a specific set of dimensionless parameters can be related to a variety of physical configurations. Then, the normalized coefficients are i ∂u ∂z + Cd 2 $_{x}$cos p ( z ) ∂ $^{2}$u ∂x 2 + D 2 ∂ $^{2}$u ∂t 2 + γ | u | $^{2}$u = 0 . (6) x$_{0}$ ( z ) = x$_{0}$$_{m}$ cos(2 πz/L ) , (7) z$_{s}$ = L$_{d}$ 2 π , t$_{s}$ = √ | D$_{d}$ | z$_{s}$, x$_{s}$ = d$_{d}$ √ 2 C$_{d}$z$_{s}$, I$_{s}$ = 1 γ$_{d}$z$_{s}$ . (8) p ( z ) = 2 πn$_{0}$dx 2 s λ$_{d}$z$_{s}$ x ′ 0 (3) a$_{n}$ = A$_{n}$/ √ I$_{s}$, z = z$_{d}$/z$_{s}$, t = t$_{d}$/t$_{s}$, C = C$_{d}$z$_{s}$, D = D$_{d}$z$_{s}$t − 2 s , γ = γ$_{d}$I$_{s}$z$_{s}$, (2) L = 2 π, d$_{x}$ = 1 2 C $_{2}$, D = σ$_{D}$ = sign( D$_{d}$ ) = ± 1 , γ = σ$_{γ}$ = sign( γ$_{d}$ ) = ± 1 , (9) i ∂u ∂z + λ ( z ) 2 ∂ $^{2}$u ∂x 2 + σ$_{D}$ 2 ∂ $^{2}$u ∂t 2 + σ$_{γ}$ | u | $^{2}$u = 0 , (10) λ ( z ) = cos( B sin z ) , B = − x$_{0}$$_{m}$ 4 π $^{2}$n$_{0}$ d$_{x}$x 2 s λ$_{d}$z$_{s}$ . (11) and Eq. (6) can be written as In self-focusing media we have σ$_{γ}$ = 1, whereas σ$_{γ}$ = − 1 in case of self-defocusing nonlinearity. In order to achieve soliton localization in temporal dimension we require that the temporal dispersion is either anomalous in self- focusing media [2], or dispersion is normal if the nonlin- earity is self-defocusing, i.e. σ$_{D}$σ$_{γ}$ > 0. As outlined in the introduction, our aim is to iden- tify the conditions for the existence of stable and mo- bile solutions in the form of light bullets, which remain localized in space and time as they propagate along the waveguide array. It was found that the solitons which are strongly localized in the spatial domain tend to become trapped at a particular lattice site [26, 27]. Only when the soliton width extends over five or more waveguides, the soliton becomes mobile [28]. However, such broad spatio-temporal solitons were found to be unstable [3, 4]. In order to study the key effects of periodic waveguide bending on the stability of broad spatio-temporal light bullets, we employ the continuous approximation. Specif- ically, we consider wavepackets that are slowly varying in space, and describe their profiles with a continuous func- tion, a$_{n}$ ( t, z ) = ˜ u ( x = nd, t, z ). We expand ˜ u in Taylor series in x and substitute in Eq. (1), where B. Continuous approximation for mobile solitons We note that there is just one parameter, the cou- pling coefficient C , which defines the mapping of solu- tions based on the continuous Eq. (10) to the original discrete-continuous Eq. (1). Continuous solutions may serve as a good approximation provided that their spa- tial width ( W ) is much larger than the distance between the neighboring waveguides, where we introduce the notation L$_{c}$ = π/C$_{d}$ to define the physical coupling length between two straight waveg- uides [29]. It follows that Eq. (10) can be used as an ap- proximate model for Eq. (1) provided that the normalized coupling coefficient ( C ), or equivalently the ratio of the bending period to the coupling length, are sufficiently large. Then, we make a transformation W ≫ d$_{x}$ = 1 2 C 2 = 2 π 2 C 2 $_{d}$L 2 d = 2 L 2 c L 2 d , (12) i ∂ ˜ u ∂z + 2 C [( ˜ u + d 2 2 ∂ $^{2}$˜ u ∂x 2 ) cos p ( z ) − id ∂ ˜ u ∂x sin p ( z ) ] + + D 2 ∂ $^{2}$˜ u ∂t 2 + γ | ˜ u | $^{2}$˜ u = 0 . (4) ˜ u ( x, t, z ) = u ( x − x$_{1}$ ( z ) , z ) exp [ 2 iC ∫ z 0 cos p ( z$_{1}$ ) dz$_{1}$ ] , (5) x$_{1}$ ( z ) = 2 Cd ∫ z 0 sin p ( z$_{1}$ ) dz$_{1}$, where