Spatiotemporal accessible solitons in fractional dimensions

Symmetry-breaking transitions in quiescent and moving solitons in fractional couplers

We consider phase transitions, in the form of spontaneous symmetry breaking (SSB) bifurcations of solitons, in dual-core couplers with fractional diffraction and cubic self-focusing acting in each core, characterized by Lévy index α. The system represents linearly coupled optical waveguides with the fractional paraxial diffraction or group-velocity dispersion (the latter system was used in a recent experiment [Nat. Commun. 14, 222 (2023)10.1038/s41467-023-35892-8], which demonstrated the first observation of the wave propagation in an effectively fractional setup). By dint of numerical computations and variational approximation, we identify the SSB in the fractional coupler as the bifurcation of the subcritical type (i.e., the symmetry-breaking phase transition of the first kind), whose subcriticality becomes stronger with the increase of fractionality 2-α, in comparison with very weak subcriticality in the case of the nonfractional diffraction, α=2. In the Cauchy limit of α→1, it carries over into the extreme subcritical bifurcation, manifesting backward-going branches of asymmetric solitons which never turn forward. The analysis of the SSB bifurcation is extended for moving (tilted) solitons, which is a nontrivial problem because the fractional diffraction does not admit Galilean invariance. Collisions between moving solitons are studied too, featuring a two-soliton symmetry-breaking effect and merger of the solitons.

Dynamics of fractional N-soliton solutions with anomalous dispersions of integrable fractional higher-order nonlinear Schrödinger equations

In this paper, using the algorithm due to Ablowitz et al. [Phys. Rev. Lett. 128, 184101 (2022); J. Phys. A: Math. Gen. 55, 384010 (2022)], we explore the anomalous dispersive relations, inverse scattering transform, and fractional N-soliton solutions of the integrable fractional higher-order nonlinear Schrödinger (fHONLS) equations, containing the fractional third-order NLS (fTONLS), fractional complex mKdV (fcmKdV), and fractional fourth-order nonlinear Schrödinger (fFONLS) equations, etc. The inverse scattering problem can be solved exactly by means of the matrix Riemann-Hilbert problem with simple poles. As a consequence, an explicit formula is found for the fractional N-soliton solutions of the fHONLS equations in the reflectionless case. In particular, we analyze the fractional one-, two-, and three-soliton solutions with anomalous dispersions of fTONLS and fcmKdV equations. The wave, group, and phase velocities of these envelope fractional one-soliton solutions are related to the power laws of their amplitudes. Moreover, we also deduce the formula for the fractional N-soliton solutions of all fHONLS equations and analyze some velocities of the one-soliton solution. These obtained fractional N-soliton solutions may be useful to explain the related super-dispersion transports of nonlinear waves in fractional nonlinear media.

Domain walls in fractional media

Currently, much interest is drawn to the analysis of optical and matter-wave modes supported by the fractional diffraction in nonlinear media. We predict a new type of such states in the form of domain walls (DWs) in the two-component system of immiscible fields. Numerical study of the underlying system of fractional nonlinear Schrödinger equations demonstrates the existence and stability of DWs at all values of the respective Lévy index (α<2), which determines the fractional diffraction, and at all values of the XPM/SPM ratio β in the two-component system above the immiscibility threshold. The same conclusion is obtained for DWs in the system which includes the linear coupling, alongside the XPM interaction between the immiscible components. Analytical results are produced for the scaling of the DW's width. The DW solutions are essentially simplified in the special case of β=3, as well as close to the immiscibility threshold. In addition to symmetric DWs, asymmetric ones are constructed too, in the system with unequal diffraction coefficients and/or different Lévy indices of the two components.

Fractional Integrable Nonlinear Soliton Equations

Nonlinear integrable equations serve as a foundation for nonlinear dynamics, and fractional equations are well known in anomalous diffusion. We connect these two fields by presenting the discovery of a new class of integrable fractional nonlinear evolution equations describing dispersive transport in fractional media. These equations can be constructed from nonlinear integrable equations using a widely generalizable mathematical process utilizing completeness relations, dispersion relations, and inverse scattering transform techniques. As examples, this general method is used to characterize fractional extensions to two physically relevant, pervasive integrable nonlinear equations: the Korteweg-deVries and nonlinear Schrödinger equations. These equations are shown to predict superdispersive transport of nondissipative solitons in fractional media.

Multiple lump and rogue wave for time fractional resonant nonlinear Schrödinger equation under parabolic law with weak nonlocal nonlinearity

This article retrieve lump, lump with one kink and rogue wave soliton for the time fractional resonant nonlinear Schrödinger equation with parabolic law having weak nonlocal nonlinearity. According to theory of dynamical systems, Schrödinger equation may be converted into plane systems. We use Hirota bilinear method to obtained these solutions. At the end, we present graphical representation of our results in various dimensions.

Generation of random soliton-like beams in a nonlinear fractional Schrödinger equation

We investigate the generation of random soliton-like beams based on the Kuznetsov-Ma solitons in a nonlinear fractional Schrödinger equation (NLFSE). For Lévy index α = 1, the Kuznetsov-Ma solitons split into two nondiffracting beams during propagation in linear regime. According to the different input positions of the Kuznetsov-Ma solitons, the diffraction-free beams can be divided into three different types: bright-dark, dark-bright and bright-bright beams. In the nonlinear regime, the Kuznetsov-Ma solitons can be evolved into random soliton-like beams due to the collapse. The number of soliton-like beams is related to the nonlinear coefficient and the Lévy index. The bigger the nonlinear coefficient, the more beams generated. Moreover, the peak intensity of soliton-like beams presents a Gaussian distribution under the large nonlinear effect. In practice, the evolution of KM soliton can be realized by a plane wave with a Gaussian perturbation, which can be confirmed that they have the similar dynamics of propagation. In two dimensions, the plane wave with a Gaussian perturbation can be evolved into a bright-dark axisymmetric ring beam in the linear regime. Under the nonlinear modulation, the energy accumulates to the center and finally breaks apart into random beam filaments.

Optical Solitons and Vortices in Fractional Media: A Mini-Review of Recent Results

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.

Symmetry-breaking bifurcations and ghost states in the fractional nonlinear Schrödinger equation with a PT-symmetric potential

We report symmetry-breaking and restoring bifurcations of solitons in a fractional Schrödinger equation with cubic or cubic-quintic (CQ) nonlinearity and a parity-time-symmetric potential, which may be realized in optical cavities. Solitons are destabilized at the bifurcation point, and, in the case of CQ nonlinearity, the stability is restored by an inverse bifurcation. Two mutually conjugate branches of ghost states (GSs), with complex propagation constants, are created by the bifurcation, solely in the case of fractional diffraction. While GSs are not true solutions, direct simulations confirm that their shapes and results of their stability analysis provide a "blueprint" for the evolution of genuine localized modes in the system.

Existence, symmetry breaking bifurcation and stability of two-dimensional optical solitons supported by fractional diffraction

We study existence, bifurcation and stability of two-dimensional optical solitons in the framework of fractional nonlinear Schrödinger equation, characterized by its Lévy index, with self-focusing and self-defocusing saturable nonlinearities. We demonstrate that the fractional diffraction system with different Lévy indexes, combined with saturable nonlinearity, supports two-dimensional symmetric, antisymmetric and asymmetric solitons, where the asymmetric solitons emerge by way of symmetry breaking bifurcation. Different scenarios of bifurcations emerge with the change of stability: the branches of asymmetric solitons split off the branches of unstable symmetric solitons with the increase of soliton power and form a supercritical type bifurcation for self-focusing saturable nonlinearity; the branches of asymmetric solitons bifurcates from the branches of unstable antisymmetric solitons for self-defocusing saturable nonlinearity, featuring a convex shape of the bifurcation loops: an antisymmetric soliton loses its stability via a supercritical bifurcation, which is followed by a reverse bifurcation that restores the stability of the symmetric soliton. Furthermore, we found a scheme of restoration or destruction the symmetry of the antisymmetric solitons by controlling the fractional diffraction in the case of self-defocusing saturable nonlinearity.

Metastable soliton necklaces supported by fractional diffraction and competing nonlinearities

We demonstrate that the fractional cubic-quintic nonlinear Schrödinger equation, characterized by its Lévy index, maintains ring-shaped soliton clusters ("necklaces") carrying orbital angular momentum. They can be built, in the respective optical setting, as circular chains of fundamental solitons linked by a vortical phase field. We predict semi-analytically that the metastable necklace-shaped clusters persist, corresponding to a local minimum of an effective potential of interaction between adjacent solitons in the cluster. Systematic simulations corroborate that the clusters stay robust over extremely large propagation distances, even in the presence of strong random perturbations.

Spontaneous symmetry breaking in purely nonlinear fractional systems

Spontaneous symmetry breaking, a spontaneous course of breaking the spatial symmetry (parity) of the system, is known to exist in many branches of physics, including condensed-matter physics, high-energy physics, nonlinear optics, and Bose-Einstein condensates. In recent years, the spontaneous symmetry breaking of solitons in nonlinear wave systems is broadly studied; understanding such a phenomenon in nonlinear fractional quantum mechanics with space fractional derivatives (the purely nonlinear fractional systems whose fundamental properties are governed by a nonlinear fractional Schrödinger equation), however, remains pending. Here, we survey symmetry breaking of solitons in fractional systems (with the fractional diffraction order being formulated by the Lévy index α) of a nonlinear double-well structure and find several kinds of soliton families in the forms of symmetric and anti-symmetric soliton states as well as asymmetric states. Linear stability and dynamical properties of these soliton states are explored relying on linear-stability analysis and direct perturbed simulations, with which the existence and stability regions of all the soliton families in the respective physical parameter space are identified.

Nonlocal solitons in fractional dimensions

We report the existence and stability properties of multipole-mode solitons supported by the nonlinear Schrödinger equation featuring a combination of the fractional-order diffraction effect and nonlocal focusing Kerr-type nonlinearity. We reveal that multipole-mode solitons, including an arbitrary number of peaks, can propagate stably in fractional systems provided that the propagation constant exceeds a certain value, which is in sharp contrast to conventional nonlocal systems under a normal diffraction, where bound states composed of five peaks or more are completely unstable. Thus, we demonstrate, to the best of our knowledge, the first example of nonlocal solitons in fractional configurations.

Anomalous interaction of Airy beams in the fractional nonlinear Schrödinger equation

We investigate the mutual interaction of two spatially-separated Airy beams in the nonlinear Schrödinger equation with the fractional Laplacian. Depending on the beam separation (d), relative phase and Lévy index (α), we observed an anomalous attraction or repulsion between the Airy beams. Anomalous attraction leads to a single breather soliton with a period that grows exponentially as α increases. In this region of the parameter space, we identify a crossover between two asymmetric regimes: as the Lévy index exceeds a critical value α _c, the period of breather soliton for d>0 is orders of magnitude larger than for d<0, while the opposite occurs as α<α _c. Our results reveal a novel scenario for Airy beams interaction in the framework of fractional nonlinear Schrödinger equation and provide an alternative mechanism to control breather soliton generation.

Optical solitons, self-focusing, and wave collapse in a space-fractional Schrödinger equation with a Kerr-type nonlinearity

We investigate the nonlinear dynamics of (1+1)-dimensional optical beam in the system described by the space-fractional Schrödinger equation with the Kerr nonlinearity. Using the variational method, the analytical soliton solutions are obtained for different values of the fractional Lévy index α. All solitons are demonstrated to be stable for 1<α≤2. However, when α=1, the beam undergoes a catastrophic collapse (blow-up) like its counterpart in the (1+2)-dimensional system at α=2. The collapse distance is analytically obtained and a physical explanation for the collapse is given.

Composition Relation between Nonlinear Bloch Waves and Gap Solitons in Periodic Fractional Systems

Evolution of beams in nonlinear optical media with a fractional-order diffraction is currently attracting a growing interest. We address the existence of linear and nonlinear Bloch waves in fractional systems with a periodic potential. Under a defocusing nonlinearity, nonlinear Bloch waves at the centers or edges of the first Brillouin zone bifurcate from the corresponding linear Bloch modes at different band edges. They can be constructed by directly copying a fundamental gap soliton (in one lattice site) or alternatively copying it and its mirror image to infinite lattice channels. The localized truncated-Bloch-wave solitons bridging nonlinear Bloch waves and gap solitons are also revealed. We thus prove that fundamental gap solitons can be used as unit cells to build nonlinear Bloch waves or truncated-Bloch-wave solitons, even in fractional configurations. Our results provide helpful hints for understanding the dynamics of localized and delocalized nonlinear modes and the relation between them in periodic fractional systems with an optical nonlinearity.

Surface gap solitons in a nonlinear fractional Schrödinger equation

We address the propagation dynamics of gap solitons at the interface between uniform media and an optical lattice in the framework of a nonlinear fractional Schrödinger equation. Different families of solitons residing in the first and second bandgaps of the Floquet-Bloch spectrum are revealed. They feature a combination of the unique properties of fractional diffraction effects, surface waves and gap solitons. The instability of solitons can be remarkably suppressed by the decrease of Lévy index, especially obvious for solitons in the second gaps. Additionally, we study the properties of multi-peaked solitons in fractional dimensions and find that they can be made completely stable in a wide region, provided that their power exceeds a critical value. Counterintuitively, at a small Lévy index, the instability region shrinks with the increase of the number of soliton peaks.

Beam propagation management in a fractional Schrödinger equation

Generalization of Fractional Schrödinger equation (FSE) into optics is fundamentally important, since optics usually provides a fertile ground where FSE-related phenomena can be effectively observed. Beam propagation management is a topic of considerable interest in the field of optics. Here, we put forward a simple scheme for the realization of propagation management of light beams by introducing a double-barrier potential into the FSE. Transmission, partial transmission/reflection, and total reflection of light fields can be controlled by varying the potential depth. Oblique input beams with arbitrary distributions obey the same propagation dynamics. Some unique properties, including strong self-healing ability, high capacity of resisting disturbance, beam reshaping, and Goos-Hänchen-like shift are revealed. Theoretical analysis results are qualitatively in agreements with the numerical findings. This work opens up new possibilities for beam management and can be generalized into other fields involving fractional effects.

Gap solitons in the nonlinear fractional Schrödinger equation with an optical lattice

We predict the existence of gap solitons in the nonlinear fractional Schrödinger equation (NLFSE) with an imprinted optically harmonic lattice. Symmetric/antisymmetric nonlinear localized modes bifurcate from the lower/upper edge of the first/second band in defocusing/focusing Kerr media. A unique feature we revealed is that, in focusing Kerr media, stable solitons appear in the finite bandgaps with the decrease of the Lévy index, which is in sharp contrast to the standard NLSE with a focusing nonlinearity. Nonlinear bound states composed by in-phase and out-of-phase soliton units supported by the NLFSE are also uncovered. Our work may pave the way for the study of spatial lattice solitons in fractional dimensions.