Gray solitons in parity-time-symmetric localized potentials with fractional-order diffraction

Highlights

• Gray solitons can be stable in the NLFSE with PT-symmetric localized potentials.

• The Lévy index will significantly affect the existence and stability of these gray solitons.

• By changing the sign of some parameters, gray solitons are changed into anti-dark solitons.

• Anti-dark solitons with fractional-order diffraction are also stable.

Abstract

The properties of existence and stability of gray solitons in parity-time (PT)-symmetric localized potentials with fractional-order diffraction are studied. This system can support stable gray solitons. The influences of the Lévy index and the real and imaginary parts of the complex potentials on existence, stability, and grayness of these solitons are carefully investigated. Especially, if some coefficients of PT-symmetric potentials in nonlinear fractional Schrödinger equation are changed, the transition between gray soliton and anti-dark soliton will occur. In addition, we also discuss the transverse energy flow in the gray and anti-dark solitons with fractional-order diffraction.

Introduction

In 2000, a promising expansion of a fractional notion in quantum physics was studied [1]. Soon, over the Lévy flight paths, the fractional path integrals were defined. In addition, both statistical mechanics and fractional quantum were discussed [2]. Later, in the semiclassical approximation, the energy spectra of fractional “Bohr atom” and oscillator were also found [3]. Moreover, in a condensed-matter system, as the probable realization of space-fractional quantum mechanics, the one-dimensional (1D) Lévy crystal was introduced and discussed [4]. The fractional Schrödinger equation (FSE) was introduced into optics in 2015 [5]. Longhi proposed an optical implementation of the FSE. Subsequently, beam propagation dynamics in the FSE were studied [6]. Distinctive properties were reported in 1D and two-dimensional (2D) cases. In the FSE, using the separation of variables method, the analytic linear mode can be found [7]. The transmission characteristics of the Airy beams in FSE were also investigated [8]. With or without a potential, evolutions of the Airy beams both exhibit unique properties. In the FSE with Gaussian and periodic potentials, the Rabi oscillations and resonant mode conversions were also reported [9]. Airy beams propagation in the FSE with the linear potential was studied [10]. The Lévy index will markedly influence propagation properties of these Airy beams. For nonlinear cases, Huang and Dong first found that gap solitons can be stable in the nonlinear FSE (NLFSE) with real periodic potentials (optical lattices) [11]. Especially, solitons in the finite gaps are stable. Later, stable surface solitons [12], 2D vortex solitons [13], [14], [15], in-phase multipole solitons [16], and vector surface solitons [17] with fractional-order diffraction were reported. In the NLFSE with nonlinear optical lattices, solitons are also stable in Kerr [18] and saturable [19] nonlinear media. The 2D linear localized potential can support stable symmetric, antisymmetric, and asymmetric solitons in NLFSE with saturable nonlinearity [20]. In the linear and nonlinear Gaussian double-well potentials, the symmetry breaking of solitons with fractional-order diffraction were also respectively reported [21], [22]. Moreover, multipole solitons in the NLFSE with nonlocal nonlinearity still remain stable [23].

The parity-time (PT)-symmetric Hamiltonian also has a real spectrum [24], [25], [26]. For a PT-symmetric potential: $V(x)$, it has such a relationship: $V(x)=V^{⁎}(-x)$. During the last decade, the PT-symmetry and the solitons supported by PT-symmetric systems were studied extensively [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47].

The combination of NLFSE and PT-symmetric potentials in solitons is a new research subject. Bright solitons in NLFSE supported by PT-symmetric localized potentials [48], [49] or optical lattices [50] were studied. We found that vector bright solitons with fractional-order diffraction are stable in PT-symmetric optical lattices [51]. In the NLFSE with defocussing Kerr nonlinearity, we reported that the in-phase multipole solitons in PT-symmetric optical lattices reside in the first finite gap and can be stable at the moderate power region [52]. In the saturable nonlinear media, we also studied the properties of solitons with fractional-order diffraction in PT-symmetric optical lattices [53]. In standard nonlinear Schrödinger equation, dark solitons in PT-symmetric complex potentials were investigated [54], [55], [56], [57]. However, dark solitons in the NLFSE with PT-symmetric localized potentials have not been addressed yet.

In this article, we discuss the properties of existence and stability of gray solitons in PT-symmetric localized potentials with fractional-order diffraction and defocussing Kerr nonlinearity. Gray solitons can be stable in this system. The Lévy index will change the grayness and existence and stability regions of the gray solitons. We also find the stable anti-dark solitons with fractional-order diffraction in the PT-symmetric localized potential. Stability analysis of these gray and anti-dark solitons is performed and it is confirmed by the soliton propagations. In addition, the transverse energy flow densities of the gray and anti-dark solitons are carefully investigated.