The Use of Higher-Order Nonlinearities: Theory

Abstract

This chapter presents basic theoretical results, which demonstrate the possibility of the existence of stable 2D and 3D solitons, both fundamental ones and solitons with embedded vorticity, under the action of the nonlinearity, which combines competing cubic self-attractive and quintic repulsive terms. 2D vortex solitons (alias vortex rings/annuli) have their stability regions for all integer values of the winding number (topological charge, alias vorticity) S = 1, 2, 3, …, while 3D vortex solitons, shaped as vortex tori (donuts), may be stable solely with S = 1. For systems of nonlinearly coupled NLS equations with the CQ nonlinearity, results are similar for vortex solitons with identical winding numbers in both components, while 2D ring-shaped solitons with opposite vorticities, ±S, in their components (hidden-vorticity solitons) are completely unstable against spontaneous splitting. Also included are results for the effective potential of interaction between far-separated 2D and 3D fundamental and vortex solitons and for the spontaneous symmetry breaking of two-component 2D spatiotemporal optical solitons in a planar dual-core coupler. The latter setting makes it possible to introduce the concept of spatiotemporal optical vortices, which feature the vortical phase distribution in the 2D plane composed of spatial and temporal coordinates [Dror and Malomed, Phys. D 240, 526–541 (2011)]. For the comparison with the setting based on the CQ nonlinearity, the chapter includes some results for the 2D model with saturable self-focusing, where all 2D and 3D vortex solitons are unstable against splitting.