Multidimensional dissipative solitons and solitary vortices

Abstract: This article offers a review of results for solitons in 2D and 3D models of nonlinear dissipative media. The existence of such solitons requires to maintain two balances: between nonlinear self-focusing and linear diffraction and/or dispersion, and between loss and gain. Due to these conditions, dissipative solitons (DSs) exist not in families, but as isolated solutions. The main issue is stability of 2D and 3D DSs, especially vortical ones. First, stable 2D DSs are presented in the framework of the complex Ginzburg-Landau equation with the cubic-quintic (CQ) nonlinearity, which combines linear and quintic loss with cubic gain. In addition to fundamental (zero-vorticity) DSs, stable spiral DSs are presented too, with vorticities 1 and 2. Stable 2D solitons were also found in a system of two linearly-coupled fields, with linear gain acting in one and linear loss in the other. In this case, the cubic loss (without quintic terms) is sufficient for the stability of fundamental and vortex DSs. In addition to truly localized states, weakly localized ones are presented too, in a model with nonlinear loss without explicit gain, the losses being compensated by influx of power from infinity. Other classes of 2D models which are considered here use spatially modulated loss or gain to predict many species of robust DSs, including ones featuring complex periodically recurring metamorphoses. Stable fundamental and vortex solitons are also produced by models with a trapping or spatially periodic potential. In the latter case, 2D gap DSs are considered. Further, 2D dissipative models with spin-orbit coupling give rise to stable semi-vortices, with vorticity carried by one component. Along with the 2D solitons, the review includes 3D fundamental and vortex DSs, stabilized by the CQ nonlinearity and/or external potentials, and collisions between them.