Basic Theoretical Models
Authors: Malomed Boris A. 
Published: September 2022

This chapter introduces most essential physical models used in the book, with the objective to stabilize fundamental 2D and 3D solitons as well as ones with embedded vorticity. The first class of models that offer this possibility is based on NLS equations in which the collapse in an optical medium, driven by the Kerr (cubic) self-attraction, is arrested by the quintic self-repulsion or by saturation of the self-attractive nonlinearity (theoretical and experimental results for models of this type are presented in Chaps. 3 and 4, respectively). Next, models are introduced with the cubic self-attraction, in which the stabilization is provided by external potentials: either trapping potentials of the harmonic-oscillator type (theoretical results for them are reported in Chap. 5), or spatially periodic (lattice) potentials. Theoretical and experimental findings for multidimensional solitons stabilized by the periodic potentials are reported, severally, in Chaps. 7 and 8. Then, models based on NLS/GP equations with the local strength of the cubic repulsive nonlinearity growing sufficiently fast from the center to periphery are presented. In that case, the stabilization of various species of multidimensional solitons, including ones carrying topological structures, is provided by an effective nonlinear potential. Theoretical predictions for stable solitons of the latter type are reported in Chap. 6. A very important setup admitting the existence of stable multidimensional soliton-like states in the form of “quantum droplets” (QDs), both fundamental ones and QDs with embedded vorticity, relies on the effective quartic repulsive nonlinearity induced by quantum fluctuations around mean-field (MF) states (the Lee–Huang–Yang effect) in binary BECs with the MF cubic attraction between the components. Theoretical results on this topic are presented in Chap. 11. It is remarkable that stable fundamental QDs in this setup have been directly demonstrated in experiments, as shown in detail in Chap. 12, and for BEC with dipole interactions between magnetic atoms is shown separately in Chap. 13. Finally, the present chapter introduces models that make it possible to predict absolutely stable 2D solitons, and metastable 3D ones, in the form of “semi-vortices” (SVs) and “mixed modes” (MMs), in binary BECs with the cubic self- and cross-attraction and spin–orbit coupling (SOC) between the components. Below, detailed results for 2D and 3D solitons stabilized by the SOC are summarized in Chap. 9. Also, considered in this chapter are 2D optical systems emulating SOC and predicting stable 2D solitons in terms of spatiotemporal propagation of light in planar dual-core couplers, including ones with the PT (parity-time) symmetry. Detailed results for stable solitons in the latter case are reported in Chap. 10. Some other models, which are considered in this book too, but in more specific contexts, are introduced later, in the framework of chapters in which they appear. In particular, these are systems with nonlocal interactions (addressed in Chap. 14) and dissipative models based on complex Ginzburg–Landau (CGL) equations (Chap. 15).