Solitons in the fractional Schrödinger equation with parity-time-symmetric lattice potential

Abstract

We investigate the properties of spatial solitons in the fractional Schrödinger equation (FSE) with parity-time (PT)-symmetric lattice potential supported by the focusing of Kerr nonlinearity. Both one-and two-dimensional solitons can stably propagate in PT-symmetric lattices under noise perturbations. The domains of stability for both one-and two-dimensional solitons strongly depend on the gain/loss strength of the lattice. In the spatial domain, the solitons are rigidly modulated by the lattice potential for the weak diffraction in FSE systems. In the inverse space, due to the periodicity of lattices, the spectra of solitons experience sharp peaks when the values of wavenumbers are even. The transverse power flows induced by the imaginary part of the lattice are also investigated, which can preserve the internal energy balances within the solitons.

Soliton dynamics in a fractional complex Ginzburg-Landau model

“…Subsequently, the propagation of beams in FSE with different external potentials and nonlinear terms was investigated [13][14][15][16][17]. In this vein, various soliton states based on FSE in Kerr nonlinear media and lattice potentials were reported recently [18][19][20][21][22][23]. In particular, it has been found that, with the decrease of LI, solitons become more localized, and their existence region essentially changes [20].…”

Quantum information entropies of multiple quantum well systems in fractional Schrödinger equations

“…However, its application to quantum mechanics was started by the pioneering work of the Laskin and others . Based on this outstanding idea, intense investigations have been carried out in different fields of studies, for example, energy band structure for the periodic potential, position‐dependent mass fractional Schrödinger equation, fractional quantum oscillator, nuclear dynamics of the H 2 + molecular ion, propagation dynamics of a light beam, spatial soliton propagation, solitons in the fractional Schrödinger equation with parity‐time‐symmetric lattice potential, gap solitons, Rabi oscillations in a fractional Schrödinger equation, self‐focusing, and wave collapse, elliptic solitons, light propagation in honeycomb lattice, and so on. These studies are based on the different methods such as domain decomposition method, energy conservative difference scheme, conservative finite element method, fractional Fan subequation method, split‐step Fourier spectral method, transfer‐matrix method, and so on.…”

Preventing critical collapse of higher-order solitons by tailoring unconventional optical diffraction and nonlinearities

“…Importantly, a new hallmark of FSE development published in 2015 by Longhi 52 found that a fractional quantum harmonic oscillator can be realized based on transverse laser beam dynamics in aspherical optical cavities, pushing the studies of fractional physical system into optics. Such work triggers a growing prosperity of fractional models for digging deep into the propagation properties of the possible beam solutions in both the linear and nonlinear systems, striking examples include: Gaussian beams propagation 53,54 , conical diffraction of light (PT symmetry) in FSE with a periodic PT -symmetric potential 55 , accessible solitons in FSE with a harmonic potential 56 , solitons 57,58 , and modulational instability 59 in purely nonlinear fractional Schrödinger equation (NLFSE), and diverse soliton families in the NLFSE setting by adding to its linear [60][61][62][63] or nonlinear 20 part with a periodic potential (the aforementioned optical and nonlinear lattices, respectively). Although the NLFSE provides a fertile ground for exploring various types of solitons, their existence and stability property supported by the cubic-quintic nonlinearities and 2D linear periodic potential (optical lattice) are yet to be revealed.…”