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The idea of fractional derivatives has a long history that dates back centuries. Apart from their intriguing mathematical properties, fractional derivatives have been studied widely in physics, for example in quantum mechanics and generally in systems with nonlocal temporal or spatial interactions. However, systematic experiments have been rare due to challenges associated with the physical implementation. Here we report the observation and full characterization of a family of temporal optical solitons that are governed by a nonlinear wave equation with a fractional Laplacian. This equation has solutions with unique properties such as non-exponential tails and a very small time-bandwidth product.

Large complex networks are often organized into groups or communities. In this paper, we introduce and investigate a generative model of network evolution that reproduces all four pairwise community types that exist in directed networks: assortative, core-periphery, disassortative, and the newly introduced source-basin type. We fix the number of nodes and the community membership of each node, allowing node connectivity to change through rewiring mechanisms that depend on the community membership of the involved nodes. We determine the dependence of the community relationship on the model parameters using a mean-field solution. It reveals that a difference in the swap probabilities of the two communities is a necessary condition to obtain a core-periphery relationship and that a difference in the average in-degree of the communities is a necessary condition for a source-basin relationship. More generally, our analysis reveals multiple possible scenarios for the transition between the different structure types, and sheds light on the mechanisms underlying the observation of the different types of communities in network data., Comment: 13 pages, 6 figures

Clustering short text is a difficult problem, due to the low word co-occurrence between short text documents. This work shows that large language models (LLMs) can overcome the limitations of traditional clustering approaches by generating embeddings that capture the semantic nuances of short text. In this study clusters are found in the embedding space using Gaussian Mixture Modelling (GMM). The resulting clusters are found to be more distinctive and more human-interpretable than clusters produced using the popular methods of doc2vec and Latent Dirichlet Allocation (LDA). The success of the clustering approach is quantified using human reviewers and through the use of a generative LLM. The generative LLM shows good agreement with the human reviewers, and is suggested as a means to bridge the `validation gap' which often exists between cluster production and cluster interpretation. The comparison between LLM-coding and human-coding reveals intrinsic biases in each, challenging the conventional reliance on human coding as the definitive standard for cluster validation., Comment: Main text: 18 pages, 6 figures. Supplementary: 21 pages, 15 figures, 3 tables

Although most soliton research has traditionally considered dominant quadratic dispersion, the recent discovery of pure-quartic solitons has inspired analysis of soliton solutions with large higher orders of dispersion. Here we present analytic expressions for families of bright soliton solutions at arbitrary dispersion orders and practical methods to obtain the associated dispersion relations. These results provide a framework for considering higher order dispersion solitons and show the potential for further investigation of solitons in higher order dispersion systems., Comment: 23 pages, 8 figures

We show theoretically that dark solitons can exist in the presence of pure quartic dispersion, and also in the presence of both quadratic and quartic dispersive effects, displaying a much greater variety of possible solutions and dynamics than for pure quadratic dispersion. The interplay of the two dispersion orders may lead to oscillatory non-vanishing tails, which enables the possibility of bound, potentially stable, multi-soliton states. Dark soliton-like states which connect to low amplitude oscillations are also shown to be possible. Dynamical evolution results corroborate the stability picture obtained, and possible avenues for dark soliton generation are explored., Comment: 4 pages, 5 figures

We consider the motion of a light sail that is accelerated by a powerful laser beam. We derive the equations of motion for two proof-of-concept sail designs with damped internal degrees of freedom. Using linear stability analysis we show that perturbations of the sail movement in all lateral degrees of freedom can be damped passively. This analysis also shows complicated behaviour akin to that associated with exceptional points in PT-symmetric systems in optics and quantum mechanics. The excess heat that is produced by the damping mechanism is likely to be substantially smaller than the expected heating due to the partial absorption of the incident laser beam by the sail., Comment: 7 figures

Temporal solitons are optical pulses that arise from the balance of negative group-velocity dispersion and self-phase modulation. For decades only quadratic dispersion was considered, with higher order dispersion thought of as a nuisance. Following the recent reporting of pure-quartic solitons, we here provide experimental and numerical evidence for an infinite hierarchy of solitons that balance self-phase modulation and arbitrary negative pure, even-order dispersion. Specifically, we experimentally demonstrate the existence of solitons with pure-sextic ($\beta_6$), -octic ($\beta_8$) and -decic ($\beta_{10}$) dispersion, limited only by the performance of our components, and show numerical evidence for the existence of solitons involving pure $16^{\rm th}$ order dispersion. Phase-resolved temporal and spectral characterization reveals that these pulses, exhibit increasing spectral flatness with dispersion order. The measured energy-width scaling laws suggest dramatic advantages for ultrashort pulses. These results broaden the fundamental understanding of solitons and present new avenues to engineer ultrafast pulses in nonlinear optics and its applications., Comment: 7 pages, 5 figures, 1 table

Soliton molecules, bound states composed of interacting fundamental solitons, exhibit remarkable resemblance with chemical compounds and phenomena in quantum mechanics. Whereas optical molecules composed of two or more temporally locked solitons have been observed in a variety of platforms, soliton molecules formed by bound solitons at different frequencies have only recently been theoretically proposed. Here, we report the observation of polychromatic soliton molecules within a mode-locked laser cavity, achieving the desired dispersion by implementing a spectral pulse-shaper. This system supports two or more coincident solitons with different frequencies, but common group velocities. Our results open new directions of exploration in nonlinear dynamics within systems with complex dispersion and offer a convenient platform for optical analogies to mutual trapping and spectral tunneling in quantum mechanics.

In this paper we quantify the statistical properties and dynamics of the frequency of hashtag use on Twitter. Hashtags are special words used in social media to attract attention and to organize content. Looking at the collection of all hashtags used in a period of time, we identify the scaling laws underpinning the hashtag frequency distribution (Zipf's law), the number of unique hashtags as a function of sample size (Heaps' law), and the fluctuations around expected values (Taylor's law). While these scaling laws appear to be universal, in the sense that similar exponents are observed irrespective of when the sample is gathered, the volume and nature of the hashtags depends strongly on time, with the appearance of bursts at the minute scale, fat-tailed noise, and long-range correlations. We quantify this dynamics by computing the Jensen-Shannon divergence between hashtag distributions obtained $\tau$ times apart and we find that the speed of change decays roughly as $1/\tau$. Our findings are based on the analysis of 3.5 billion hashtags used between 2015 and 2016., Comment: 8 pages and 4 figures. Submitted to the journal "Chaos", special edition on "Dynamics of Social Systems". Data available at https://zenodo.org/record/3673744#.Xqa5t_GhSv4

We study self-trapped localized nonlinear states in the form of truncated Bloch waves in one-dimensional optical lattices, which appear in the gaps of the linear bandgap spectrum. We demonstrate the existence of families of such localized states which differ by the number of intensity peaks. These families do not bifurcate from the band edge, and their power curves exhibit double branches. Linear stability analysis demonstrates that in deep lattice potentials the states corresponding to the lower branches are stable, whereas those corresponding to the upper branches are unstable, independently of the number of peaks., Comment: To appear in Phys. Rev. A

We analyze the structure of spin-1 Bose-Einstein condensates in the presence of a homogenous magnetic field. We classify the homogenous stationary states and study their existence, bifurcations, and energy spectra. We reveal that the phase separation can occur in the ground state of polar condensates, while the spin components of the ferromagnetic condensates are always miscible and no phase separation occurs. Our theoretical model, confirmed by numerical simulations, explains that this phenomenon takes place when the energy of the lowest homogenous state is a concave function of the magnetization. In particular, we predict that phase separation can be observed in a $^{23}$Na condensate confined in a highly elongated harmonic trap. Finally, we discuss the phenomena of dynamical instability and spin domain formation.

We show that double-charge discrete optical vortices may be completely stable in hexagonal photonic lattices where single-charge vortices always exhibit dynamical instabilities. Even when unstable the double-charge vortices typically have a much weaker instability than the single-charge vortices, and thus their breakup occurs at longer propagation distances.

Antiferromagnetic condensates are generally believed not to display modulational instability and subsequent spin-domain formation. Here we demonstrate that in he presence of a homogeneous magnetic field antiferromagnetic spin-1 Bose-Einstein condensates can undergo spatial modulational instability followed by the subsequent generation of spin domains. Employing numerical simulations for realistic conditions, we show how this novel effect can be observed in sodium condensates confined in an optical trap. Finally, we link this instability and spin-domain formation with stationary modes of the condensate.

We study the dynamics of bright solitons formed in a Bose-Einstein condensate with attractive atomic interactions perturbed by a weak bichromatic optical lattice potential. The lattice depth is a biperiodic function of time with a zero mean, which realises a flashing ratchet for matter-wave solitons. The average velocity of a soliton and the directed soliton current induced by the ratchet depend on the number of atoms in the soliton. We employ this feature to study collisions between ratchet-driven solitons and find that soliton transport can be induced through their interactions. In the regime when matter-wave solitons are narrow compared to the lattice period the ratchet dynamics is well described by the effective Hamiltonian theory., Comment: 4 pages, 5 figures

We model the nonlinear behaviour of spin-1 Bose-Einstein condensates (BECs) with repulsive spin-independent interactions and either ferromagnetic or anti-ferromagnetic (polar) spin-dependent interactions, loaded into a one-dimensional optical lattice potential. We show that both types of BECs exhibit dynamical instabilities and may form spatially localized multi-component structures. The localized states of the spinor matter waves take the form of vector gap solitons and self-trapped waves that exist only within gaps of the linear Bloch-wave band-gap spectrum. Of special interest are the nonlinear localized states that do not exhibit a common spatial density profile shared by all condensate components, and consequently cannot be described by the single mode approximation (SMA), frequently employed within the framework of the mean-field treatment. We show that the non-SMA states can exhibits Josephson-like internal oscillations and self-magnetisation, i.e. intrinsic precession of the local spin. Finally, we demonstrate that non-stationary states of a spinor BEC in a lattice exhibit coherent undamped spin-mixing dynamics, and that their controlled conversion into a stationary state can be achieved by the application of an external magnetic field., Comment: 12 pages, 13 figures

We consider nonlinear boson states with a nontrivial phase structure in the three-site Bose-Hubbard ring, {\em quantum discrete vortices} (or {\em q-vortices}), and study their "melting" under the action of quantum fluctuations. We calculate the spatial correlations in the ground states to show the superfluid-insulator crossover and analyze the fidelity between the exact and variational ground states to explore the validity of the classical analysis. We examine the phase coherence and the effect of quantum fluctuations on q-vortices and reveal that the breakdown of these coherent structures through quantum fluctuations accompanies the superfluid-insulator crossover., Comment: Revised version, 4 pages, 5 figures, Accepted for publication in Physical Review Letters

We predict the existence of spatially localized nontrivial vortex states of a Bose-Einstein condensate with repulsive atomic interaction confined by a three-dimensional optical lattice. Such vortex-like structures include planar vortices, their co- and counter-rotating bound states, and distinctly three-dimensional non-planar vortex states. We demonstrate numerically that many of these vortex structures are remarkably robust, and therefore can be generated and observed in experiment., Comment: 4 pages, 6 figures

We reveal the existence of asymmetric vortex solitons in ideally symmetric periodic lattices, and show how such nonlinear localized structures describing elementary circular flows can be analyzed systematically using the energy-balance relations. We present the examples of rhomboid, rectangular, and triangular vortex solitons on a square lattice, and also describe novel coherent states where the populations of clockwise and anti-clockwise vortex modes change periodically due to a nonlinearity-induced momentum exchange through the lattice. Asymmetric vortex solitons are expected to exist in different nonlinear lattice systems including optically-induced photonic lattices, nonlinear photonic crystals, and Bose-Einstein condensates in optical lattices., Comment: 4 pages, 5 figures

We report on the frst experimental observation of discrete vortex solitons in two-dimensional optically-induced photonic lattices. We demonstrate strong stabilization of an optical vortex by the lattice in a self-focusing nonlinear medium and study the generation of the discrete vortices from a broad class of singular beams., Comment: 4pages, 5 colour figures. to appear in PRL

The structure and stability of vortices in hybrid atomic-molecular Bose-Einstein condensates is analyzed in the framework of a two-component Gross-Pitaevskii-type model that describes the stimulated Raman-induced photoassociation process. New types of topological vortex states are predicted to exist in the coherently coupled two-component condensates even without a trap, and their nontrivial dynamics in the presence of losses is demonstrated., Comment: 7 pages, 6 figures

We analyze the model of an atomic Bose-Einstein condensate (BEC) parametrically coupled to a molecular BEC via a photoassociation process. We show that an interplay of nonlinear inter- and intra-species interactions leads to the formation of mutually trapped states of a hybrid condensate, which are spatially localized even without a trap. The untrapped atomic-molecular condensates with a large number of atoms in either fraction are shown to be dynamically stable in the framework of a mean field theory., Comment: 4 pages, 4 figures

We apply the concept of multistep cascading to the problem of fourth-harmonic generation in a single quadratic crystal. We analyze a new model of parametric wave mixing and describe its stationary solutions for two- and three-color plane waves and spatial solitons. Some applications to the optical frequency division as well as the realization of the double-phase-matching processes in engineered QPM structures with phase reversal sequences are also discussed., Comment: 3 pages, 3 figures

We discuss several novel types of multi-component (temporal and spatial) envelope solitary waves that appear in fiber and waveguide nonlinear optics. In particular, we describe multi-channel solitary waves in bit-parallel-wavelength fiber transmission systems for high performance computer networks, multi-colour parametric spatial solitary waves due to cascaded nonlinearities of quadratic materials, and quasiperiodic envelope solitons due to quasi-phase-matching in Fibonacci optical superlattices., Comment: 12 pages, 11 figures; To be published in: Proceedings of the Dynamics Days Asia-Pacific: First International Conference on Nonlinear Science (Hong-Kong, 13-16 July, 1999), Editor: Bambi Hu (Elsevier Publishers, 2000)

We study, numerically and analytically, linear and nonlinear waveguides induced by optical vortex solitons in a Kerr medium. Both fundamental and first-order guided modes are analyzed, as well as the cases of effectively defocusing and focusing nonlinearity., Comment: 3 pages, 3 figures, changed content

We consider the Bose-Einstein condensate in a parabolic trap as a macroscopic quantum oscillator and describe, analytically and numerically, its collective modes - a nonlinear generalisation of the (symmetric and antisymmetric) Hermite-Gauss eigenmodes of a harmonic quantum oscillator., Comment: 4 pages, 2 figures, submitted to the Physical Review Revised version (minor changes)

A general stability criterion is derived for the D-dimensional ground states of the Gross-Pitaevskii equation, which describes attractive Bose-Einstein condensates confined in a magnetic trap. These ground states are shown to avoid the collapse in finite time and are proven to be stable in two and three spatial dimensions., Comment: 6 pages, 3 figures, submitted to the Physical Review Revised and extended version

We analyse systematically, from the viewpoint of the nonlinear physics of solitary waves, the effect of the spatial dimension (D = 1,2,3) on the structure and stability of the Bose-Einstein condensates (BECs) trapped in an external anisotropic parabolic potential. While for the positive scattering length the stationary ground-state solutions of the Gross-Pitaevskii equation are shown to be always stable independently of the spatial dimension, for the negative scattering length the ground-state condensate is stable only in the 1D and 2D cases, whereas in the 3D case it becomes unstable. A direct link between nonlinear modes of BECs and (bright and dark) solitary waves of the nonlinear Schrodinger equation is demonstrated for all the dimensions., Comment: 11 pages, 10 figures To appear in the Proceeding of the APCTP-Nankai Symposium on Yang-Baxter Systems, Nonlinear Models and Their Applications. Eds. Q-Han Park et al. (World Scientific, Singapore, 1999) in press

We introduce a novel class of parametric optical solitons supported simultaneously by two second-order nonlinear cascading processes, second-harmonic generation and sum-frequency mixing. We obtain, analytically and numerically, the solutions for three-wave spatial solitons and show that the presence of an additional cascading mechanism can change dramatically the properties and stability of two-wave quadratic solitary waves., Comment: 6 pages, 4 figures

We analyze two-component spatial optical vortex solitons supported by parametric wave mixing processes in a nonlinear bulk medium. We study two distinct cases of such localised waves, namely, parametric vortex solitons due to phase-matched second-harmonic generation in an optical medium with competing quadratic and cubic nonlinear response, and vortex solitons in the presence of third-harmonic generation in a cubic medium. We find, analytically and numerically, the structure of two-component vortex solitons, and also investigate modulational instability of their plane-wave background. In particular, we predict and analyze in detail novel types of vortex solitons, a `halo-vortex', consisting of a two-component vortex core surrounded by a bright ring of its harmonic field, and a `ring-vortex' soliton which is a vortex in a harmonic field that guides a ring-like localized mode of the fundamental-frequency field., Comment: 16 pages, 10 figures Revised version