Abstract
The article produces a brief review of some recent results which predict stable propagation of solitons and solitary vortices in models based on the nonlinear Schrödinger equation (NLSE) including fractional one-dimensional or two-dimensional diffraction and cubic or cubic-quintic nonlinear terms, as well as linear potentials. The fractional diffraction is represented by fractional-order spatial derivatives of the Riesz type, defined in terms of the direct and inverse Fourier transform. In this form, it can be realized by spatial-domain light propagation in optical setups with a specially devised combination of mirrors, lenses, and phase masks. The results presented in the article were chiefly obtained in a numerical form. Some analytical findings are included too, in particular, for fast moving solitons and the results produced by the variational approximation. Moreover, dissipative solitons are briefly considered, which are governed by the fractional complex Ginzburg–Landau equation.
Highlights
Introduction and the Basic ModelsNonlinear Schrödinger equations (NLSEs) give rise to soliton families in a great number of realizations [1,2,3,4,5,6,7,8], many of which originate in optics
Helps to create stable solitons even in this case [34]. This finding is similar to the wellknown fact that the parabolic trap lifts the norm degeneracy of the Townes solitons and stabilizes their entire family against the critical collapse in the framework of the usual two-dimensional NLSE with the cubic self-attraction [54,55]
The same potential makes it possible to predict the existence of stable higher-order solutions of Equation (1). Such solutions may be considered as a nonlinear extension of various excited bound states maintained by the parabolic trapping potential in the linear Schrödinger equation. The latter finding is similar to the ability of the 2D parabolic potential to partly stabilize a family of trapped vortex solitons with winding number n = 1 in the framework of the cubic self-attractive NLSE [54,55]
Summary
Introduction and the Basic ModelsNonlinear Schrödinger equations (NLSEs) give rise to soliton families in a great number of realizations [1,2,3,4,5,6,7,8] , many of which originate in optics. The modulational instability of continuous waves [17] and many types of optical solitons produced by fractional NLSEs [18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48] has been theoretically investigated, by means of numerical methods. Dissipative solitons in fractional complex Ginzburg–Landau equation (CGLE) were studied too [39]
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