Experimental realisations of the fractional Schrödinger equation in the temporal domain

Numerical investigation of the fractional-soliton mode-locked fiber laser

We propose and numerically investigate a fractional-soliton mode-locked fiber laser by utilizing an intracavity spectral pulse shaper (SPS). The fiber laser can generate stable fractional-soliton pulses for three different Lévy index α (1 < α < 2), whose profiles are all close to the sech shape. We find that the positions of Kelly sidebands, pulse energy, and peak power of the emitted fractional pulses conform to three theoretical expressions, respectively. The numerical results are in good agreement with the theoretical analyses. In addition, the intracavity dynamics of the fractional pulses have been discussed. Our findings not only deepen the fundamental understanding of temporal fractional soliton but also provide a novel, to the best of our knowledge, approach to generating stable ultrashort fractional pulses.

Modulational instability, generation, and evolution of rogue waves in the generalized fractional nonlinear Schrödinger equations with power-law nonlinearity and rational potentials

In this paper, we investigate several properties of the modulational instability (MI) and rogue waves (RWs) within the framework of the generalized fractional nonlinear Schrödinger (FNLS) equations with rational potentials. We derive the dispersion relation for a continuous wave (CW), elucidating the relationship between the wavenumber and the instability growth rate of the CW solution in the absence of potentials. This relationship is primarily influenced by the power parameter σ, the Lévy index α, and the nonlinear coefficient g. Our theoretical findings are corroborated by numerical simulations, which demonstrate that MI occurs in the focusing context. Furthermore, we study the RW generations in both cubic and quintic FNLS equations with two types of time-dependent rational potentials, which make both cubic and quintic NLS equations support the exact RW solutions. Specifically, we show that the introduction of these two potentials allows for the excitations of controllable RWs in the defocusing regime. When these two potentials become the time-independent cases such that the stable W-shaped solitons with non-zero backgrounds are generated in these cubic and quintic FNLS equations. Moreover, we consider the excitations of higher-order RWs and investigate the conditions necessary for their generations. Our analysis reveals the intricate interplay between the system parameters and the potential configurations, offering insights into the mechanisms that facilitate the emergence of higher-order RWs. Finally, we find the separated controllable multi-RWs in the defocusing cubic FNLS equation with time-dependent multi-potentials. This comprehensive study not only enhances our understanding of MI and RWs in the fractional nonlinear wave systems, but also paves the way for future research in related nonlinear wave phenomena.

Non-Markovian quantum mechanics on comb

Quantum dynamics of a particle on a two-dimensional comb structure is considered. This dynamics of a Hamiltonian system with a topologically constrained geometry leads to the non-Markovian behavior. In the framework of a rigorous analytical consideration, it is shown how a fractional time derivative appears for the relevant description of this non-Markovian quantum mechanics in the framework of fractional time Schrödinger equations. Analytical solutions for the Green functions are obtained for both conservative and periodically driven in time Hamiltonian systems.

Dynamics of discrete solitons in the fractional discrete nonlinear Schrödinger equation with the quasi-Riesz derivative

We elaborate a fractional discrete nonlinear Schrödinger (FDNLS) equation based on an appropriately modified definition of the Riesz fractional derivative, which is characterized by its Lévy index (LI). This FDNLS equation represents a novel discrete system, in which the nearest-neighbor coupling is combined with long-range interactions, that decay as the inverse square of the separation between lattice sites. The system may be realized as an array of parallel quasi-one-dimensional Bose-Einstein condensates composed of atoms or small molecules carrying, respectively, a permanent magnetic or electric dipole moment. The dispersion relation (DR) for lattice waves and the corresponding propagation band in the system's linear spectrum are found in an exact form for all values of LI. The DR is consistent with the continuum limit, differing in the range of wave numbers. Formation of single-site and two-site discrete solitons is explored, starting from the anticontinuum limit and continuing the analysis in the numerical form up to the existence boundary of the discrete solitons. Stability of the solitons is identified in terms of eigenvalues for small perturbations, and verified in direct simulations. Mobility of the discrete solitons is considered too, by means of an estimate of the system's Peierls-Nabarro potential barrier, and with the help of direct simulations. Collisions between persistently moving discrete solitons are also studied.

Deciphering and integrating invariants for neural operator learning with various physical mechanisms

Neural operators have been explored as surrogate models for simulating physical systems to overcome the limitations of traditional partial differential equation (PDE) solvers. However, most existing operator learning methods assume that the data originate from a single physical mechanism, limiting their applicability and performance in more realistic scenarios. To this end, we propose the physical invariant attention neural operator (PIANO) to decipher and integrate the physical invariants for operator learning from the PDE series with various physical mechanisms. PIANO employs self-supervised learning to extract physical knowledge and attention mechanisms to integrate them into dynamic convolutional layers. Compared to existing techniques, PIANO can reduce the relative error by 13.6%-82.2% on PDE forecasting tasks across varying coefficients, forces or boundary conditions. Additionally, varied downstream tasks reveal that the PI embeddings deciphered by PIANO align well with the underlying invariants in the PDE systems, verifying the physical significance of PIANO.

Basic fractional nonlinear-wave models and solitons

This review article provides a concise summary of one- and two-dimensional models for the propagation of linear and nonlinear waves in fractional media. The basic models, which originate from Laskin's fractional quantum mechanics and more experimentally relevant setups emulating fractional diffraction in optics, are based on the Riesz definition of fractional derivatives, which are characterized by the respective Lévy indices. Basic species of one-dimensional solitons, produced by the fractional models which include cubic or quadratic nonlinear terms, are outlined too. In particular, it is demonstrated that the variational approximation is relevant in many cases. A summary of the recently demonstrated experimental realization of the fractional group-velocity dispersion in fiber lasers is also presented.

Controlling beam dynamics with spectral quadratic phase modulation in the fractional Schrödinger equation

The propagation dynamics of Gaussian beams and finite energy Airy beams with spectral quadratic phase modulation (QPM) modeled by the fractional Schrödinger equation (FSE) are numerically investigated. Compared with beam propagation in the standard Schrödinger equation, the focusing property of beams under FSE is influenced by the QPM coefficient and the Lévy index. For symmetric Gaussian beams, the focusing position increases and the focusing intensity decreases for the larger QPM coefficient or smaller Lévy index. For asymmetric Airy beams, multiple focusing positions occur, and the tendency of focusing intensity is opposite to that of Gaussian beams. Our results show the promising application of the FSE system for optical manipulation and optical splitting by controlling the QPM.

One-dimensional Lévy quasicrystal

Space-fractional quantum mechanics (SFQM) is a generalization of the standard quantum mechanics when the Brownian trajectories in Feynman path integrals are replaced by Lévy flights. We introduce Lévy quasicrystal by discretizing the space-fractional Schrödinger equation using the Grünwald-Letnikov derivatives and adding on-site quasiperiodic potential. The discretized version of the usual Schrödinger equation maps to the Aubry-André (AA) Hamiltonian, which supports localization-delocalization transition even in one dimension. We find the similarities between Lévy quasicrystal and the AA model with power-law hopping, and show that the Lévy quasicrystal supports a delocalization-localization transition as one tunes the quasiperiodic potential strength and shows the coexistence of localized and delocalized states separated by mobility edge. Hence, a possible realization of SFQM in optical experiments should be a new experimental platform to test the predictions of AA models in the presence of power-law hopping.

Quantum Information Entropy for a Hyperbolic Double Well Potential in the Fractional Schrödinger Equation

In this study, we investigate the position and momentum Shannon entropy, denoted as Sx and Sp, respectively, in the context of the fractional Schrödinger equation (FSE) for a hyperbolic double well potential (HDWP). We explore various values of the fractional derivative represented by k in our analysis. Our findings reveal intriguing behavior concerning the localization properties of the position entropy density, ρs(x), and the momentum entropy density, ρs(p), for low-lying states. Specifically, as the fractional derivative k decreases, ρs(x) becomes more localized, whereas ρs(p) becomes more delocalized. Moreover, we observe that as the derivative k decreases, the position entropy Sx decreases, while the momentum entropy Sp increases. In particular, the sum of these entropies consistently increases with decreasing fractional derivative k. It is noteworthy that, despite the increase in position Shannon entropy Sx and the decrease in momentum Shannon entropy Sp with an increase in the depth u of the HDWP, the Beckner-Bialynicki-Birula-Mycielski (BBM) inequality relation remains satisfied. Furthermore, we examine the Fisher entropy and its dependence on the depth u of the HDWP and the fractional derivative k. Our results indicate that the Fisher entropy increases as the depth u of the HDWP is increased and the fractional derivative k is decreased.

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.