Multidimensional solitons and vortices in periodic potentials

The existence of stable solitons and localized vortices in two- and three-dimensional (2D and 3D) media governed by the cubic nonlinear Schroedinger equation with a periodic potential is demonstrated by means of the variational approximation (VA) and in direct simulations. In the 2D case, multi-mode (hexagonal, triangular, and quasi periodic) potentials are considered (including search for vortex solitons in them), along with the usual square potential. In the 2D and 3D cases, low-dimensional (respectively quasi-1D and quasi-2D) potentials are considered too. Families of solitons include single- and multi-peaked ones. Solitons of the former type and their stability are well predicted by VA. Collisions of multidimensional solitons in a low dimensional periodic potential are also studied. Head-on collisions and in-phase solitons lead to their fusion into a collapsing pulse. Soliton colliding in adjacent lattice-.induced channels may form a bound state (BS), which then relaxes to a stable asymmetric form. An initial unstable soliton splits into a three-soliton BS. The results apply to Bose-Einstein condensates (BECs) in optical lattices (OLs), and to spatial or spatiotemporal solitons in layered optical media.