Spatiotemporal accessible solitons in fractional dimensions

Abstract

Abstract：We report solutions for solitons of the "accessible" type in globally nonlocal nonlinear media of fractional dimension (FD), viz., for self-trapped modes in the space of effective dimension 3 2   D with harmonic-oscillator potential whose strength is proportional to the total power of the wave field. The solutions are categorized by a combination of radial, orbital and azimuthal quantum numbers ) , , ( m l n . They feature coaxial sets of vortical and necklace-shaped rings of different orders, and can be exactly written in terms of special functions that include Gegenbauer polynomials, associated Laguerre polynomials, and associated Legendre functions. The validity of these solutions is verified by direct simulation. The model can be realized in various physical settings emulated by FD spaces; in particular, it applies to excitons trapped in quantum wells.

One-dimensional gap solitons in quintic and cubic–quintic fractional nonlinear Schrödinger equations with a periodically modulated linear potential

“…Since then, interesting results on generating and manipulating linear and nonlinear propagation dynamics of laser beams in such fractional optical models were obtained. Some typical works include: Gaus- * zengjh@opt.ac.cn sian beams either evolved into diffraction-free beams [14] or undergone conical diffraction [15] during propagation without a potential, PT symmetry [16] and propagation dynamics of the super-Gaussian beams [17] , optical beams propagation with a harmonic potential [14,15,17] (which supports spatiotemporal accessible solitons too [18,19]) and periodic potentials [16,20], propagation management of light beams in a double-barrier potential [21], in the context of linear FSE regime; and in terms of nonlinear fractional Schrödinger equation (NLFSE) regime [22][23][24][25][26][27][28], including optical solitons (or solitary waves) without external potential [23,24], solitons supported by linear [25][26][27] and nonlinear [28] periodic potentials which refer, respectively, to optical lattice and nonlinear lattice as described below.…”

“…The purely nonlinear defocusing media with spatially inhomogeneous nonlinearity whose local strength grows quickly enough from the center toward periphery in the D-dimensional coordinate, which are built on self-defocusing background and therefore do not possess critical and supercritical collapsestypical characteristics for solitons in self-focusing media, enriched the generation of various families of stable solitons and soliton composites in the self-trapping regime [75][76][77][78][79][80][81][82][83][84][85][86][87][88], such as the fundamental solitons for all (D-dimensional) space coordiantes [75,76], 1D multihump states in forms of dipole and multipole solitons [75][76][77], 2D bright solitary vortices carrying with an arbitrarily high topological charge [75,76], 2D localized dark solitons and vortices [86], multifarious 3D localized modes that are comprised of soliton gyroscopes [81] and skyrmions [82], and very recently the flat-top solitons (in both 1D and 2D spaces) and 2D vortices [88,89], to name just some of them. Despite excellent research works on solitons in fractional Schrödinger equation are making headway in past few years [18,19,[23][24][25][26][27][28], the presence of solitons and their propagation properties in periodic potentials with quintic nonlinearity or in purely cubic-quintic model (without any external potential) mentioned above and combinations thereof are yet for investigating. In this article, we incorporate an external linear potential (optical lattice) into the 1D cubic-quintic or quinticonly NLFSE and examine the formation and propagation dynamics of localized gap modes.…”

Beam propagation management in a fractional Schrödinger equation

“…investigated the beam dynamics in the FSE with or without an external potential and found series of fascinating features, including the zigzag propagation of chirped Gaussian beam in a parabolic potential 28 , conical diffraction in -symmetry periodic lattices 29 , diffraction-free beams in uniform media 30 , and linear modes trapped in a harmonic-oscillator potential 31, 32 . Note that the experimental setting for the realization of beam propagation in the FSE was proposed very recently 30 .…”

Fraction‐Dimensional Accessible Solitons in a Parity‐Time Symmetric Potential

“…We have extended the SM model to non‐integer dimensions, and obtained some surprising results . This paper presents a further extension of these efforts in the direction of including the influence of PT‐symmetric potentials on fraction‐dimensional accessible solitons.…”