Symmetry Breaking in Fractional Nonlinear Schrödinger and Soliton Dynamics in Complex Ginzburg-Landau Models

In this chapter we review some recent results for fractional nonlinear Schrödinger (FNLS) and fractional complex Ginzburg-Landau (FCGL) models. In particular, one- and two-dimensional solitons, as well as their spontaneous symmetry breaking (SSB), have been studied in the framework of FNLS equations with real and complex \(\mathcal{P}\mathcal{T}\) -symmetric potentials by means of numerical methods. The numerical analysis was also developed for multidimensional solitons with fractional dispersion and diffraction effects. FNLS equations with real potentials can support symmetric, antisymmetric, and asymmetric solitons, the asymmetric ones emerging by way of SSB bifurcations. Different bifurcation scenarios appear with the change of stability of the symmetric and antisymmetric solitons. Power curves of the SSB bifurcations produce real propagation constants for a real double-well potential, and two mutually conjugate branches of “ghost states” with complex propagation constants, created by the SSB bifurcations for complex \(\mathcal{P}\mathcal{T}\) -symmetric potentials. We also analyze the beam dynamics governed by the FNLS and FCGL equations, reporting effects of the fractional diffraction on the evolution of solitons.