Spatially Periodic Potentials (Lattices): Theory

Abstract

The use of spatially periodic (lattice) potentials, such as the 1D, 3D, and radial ones, defined by Eqs. (1.63), (2.20), and (2.24), respectively, offers a universal framework for the stabilization of multidimensional solitons and solitary vortices, as well as bound states of solitons. The objective of this chapter is to summarize various theoretical results that demonstrate such possibilities. These findings are closely related to the topic of discrete optics, as the wave dynamics in media including sufficiently deep lattice potentials is very similar, in linear and nonlinear settings alike, to the wave propagation in arrays of discrete guiding channels (Lederer et al., 2008). In particular, the arrest of the collapse by periodic potentials makes the aborted blowup of the wave field similar to the effect of quasi-collapse which occurs in waveguiding arrays [Aceves et al., Phys. Rev. Lett. 75, 73–76 (1995)]. Particular results reported in this chapter include stabilization of 2D and 3D fundamental and vortical solitons by lattice potential with the full or reduced dimension (in particular, the 2D lattice is sufficient for the stabilization of 3D solitons), two-dimensional “supervortices” (ring-shaped chains of compact eddies with global vorticity imprinted onto the chains), 2D gap solitons, which demonstrate high mobility with an effective negative mass, 2D solitons stabilized by radial and quasi-periodic lattice potentials, and 2D vortex solitons in second-harmonic-generating media stabilized by lattice potentials.