Shapes and dynamics of semi-discrete solitons in arrayed and stacked waveguiding systems

We study the shapes and dynamics of semi-discrete solitons (SDSs) in the known model of the set of linearly-coupled waveguides with the self-focusing cubic nonlinearity. The model describes a planar array of optical fibers, as well as a stack of parallel planar waveguides, in the temporal and spatial domains, respectively. It also applies to the self-attractive Bose-Einstein condensate (BEC) loaded into an array of parallel tunnel-coupled "cigar-shaped" traps. It was known that the interplay of the group-velocity dispersion, discrete diffraction and nonlinearity gives rise to SDSs in this system. We here develop a variational approximation (VA) and additional analytical methods for the description of the SDSs, and study their mobility and collisions by means of direct simulations. The VA and another analytical method (an exact solution of the linearized equation in the core adjacent to the central one) produce an accurate description for the family of fundamental onsite-centered SDS solutions. The VA is developed too for transversely unstable intersite-centered and twisted solitons. In simulations, the solitons are not mobile in the discrete direction (while broader non-soliton modes may be mobile across the array). Collisions between the SDSs traveling in the longitudinal direction reveal a threshold separating the passage and merger, or mutual destruction of the colliding solitons. While the exact shape of the threshold, considered as a function of the solitons' energy, features irregularities, its average form is explained in an analytical form. Collisions between SDSs centered at two adjacent cores are studied too.