Multidimensional Dissipative Solitons and Solitary Vortices

Abstract

Unlike other chapters of the book, which deal with conservative (or nearly conservative) settings, this chapter addresses multidimensional dissipative solitons (DSs), i.e., self-trapped states in nonlinear media with losses and compensating gain. In this case, the existence of solitons requires to maintain two distinct balances: between nonlinear self-attraction of waves and spreading of the wave fields under the action of diffraction and GVD, and balance between the losses and gain. Due to the interplay of these conditions, DSs exist, unlike solitons in conservative and PT-symmetric media, not in continuous families, but as isolated solutions. Similar to the case of conservative systems, the main issue is the stability of multidimensional DSs, especially the ones with embedded vorticity. First, stable 2D DSs are found in the framework of the complex Ginzburg–Landau equation with the CQ (cubic-quintic) nonlinearity, which combines linear loss, cubic gain, and quintic loss (the linear loss is necessary to stabilize zero background around DSs). In addition to fundamental (zero-vorticity) solitons, stable spiral solitons are found, with vorticities S = 1 and 2. Stable 2D solitons are also produced in a system built of two linearly-coupled cores, with linear gain acting in one core and linear loss, which plays the stabilizing role, in the other. In this case, the inclusion of the cubic loss (without quintic terms) is sufficient for the creation of stable fundamental and vortical DSs in the dual-core coupler. In addition to truly localized states, weakly localized ones are presented too, in the single-component model with nonlinear losses, which does not include explicit gain. In that case, the losses are compensated by the influx of power from the reservoir provided by the weakly localized structure of the solution. Other classes of 2D models which are considered in this chapter make use of spatially modulated losses or gain to predict many species of robust DSs, including those featuring complex peridically recurring metamorphoses. Stable fundamental and vortical solitons are also produced by models including a trapping or spatially periodic potential. In the latter case, the consideration addresses gap dissipative solitons as well. 2D dissipative models including spin–orbit coupling and solitons of the semi-vortex solitons in them are considered too. Stable three-dimensional fundamental and vortical DSs reported in the chapter are stabilized by the CQ nonlinearity and/or external potentials. Collisions between 3D DSs are considered at the end of the chapter.