Stability of finite and infinite von Kármán vortex-cluster streets.

A wake of vortices with sufficiently spaced cores may be represented via the point-vortex model from classical hydrodynamics. We use potential theory representations of vortices to examine the emergence and stability of complex vortex wakes, more particularly the von Kármán vortex street composed of regular polygonal-like clusters of same-signed vortices. We investigate the existence and stability of these streets represented through spatially periodic vortices. We introduce a physically inspired point-vortex model that captures the stability of infinite vortex streets with a finite number of procedurally generated vortices, allowing for numerical analysis of the behavior of vortex streets as they dynamically form.

Characterizing coherent structures in Bose-Einstein condensates through dynamic-mode decomposition.

We use the dynamic mode decomposition (DMD) methodology to study weakly turbulent flows in two-dimensional Bose-Einstein condensates modeled by a Gross-Pitaevskii equation subject to band-limited stochastic forcing. The forcing is balanced by the removal of energy at both ends of the energy spectrum through phenomenological hypoviscosity and hyperviscosity terms. Using different combinations of these parameters, we simulate three different regimes corresponding to weak-wave turbulence, and high- and low-frequency saturation. By extracting and ranking the primary DMD modes carrying the bulk of the energy, we are able to characterize the different regimes. In particular, the proposed DMD mode projection is able to seamlessly extract the vortices present in the condensate. This is achieved despite the fact that we do not use any phase information of the condensate as it is usually not directly available in realistic atomic BEC scenarios. Being model independent, this DMD methodology should be portable to other models and experiments involving complex flows. The DMD implementation could be used to elucidate different types of turbulent regimes as well as identifying and pinpointing the existence of delicate and hidden coherent structures within complex flows.

Hydrodynamics and two-dimensional dark lump solitons for polariton superfluids.

We study a two-dimensional incoherently pumped exciton-polariton condensate described by an open-dissipative Gross-Pitaevskii equation for the polariton dynamics coupled to a rate equation for the exciton density. Adopting a hydrodynamic approach, we use multiscale expansion methods to derive several models appearing in the context of shallow water waves with viscosity. In particular, we derive a Boussinesq/Benney-Luke-type equation and its far-field expansion in terms of Kadomtsev-Petviashvili-I (KP-I) equations for right- and left-going waves. From the KP-I model, we predict the existence of vorticity-free, weakly (algebraically) localized two-dimensional dark-lump solitons. We find that, in the presence of dissipation, dark lumps exhibit a lifetime three times larger than that of planar dark solitons. Direct numerical simulations show that dark lumps do exist, and their dissipative dynamics is well captured by our analytical approximation. It is also shown that lumplike and vortexlike structures can spontaneously be formed as a result of the transverse "snaking" instability of dark soliton stripes.

Adiabatic Invariant Approach to Transverse Instability: Landau Dynamics of Soliton Filaments.

Consider a lower-dimensional solitonic structure embedded in a higher-dimensional space, e.g., a 1D dark soliton embedded in 2D space, a ring dark soliton in 2D space, a spherical shell soliton in 3D space, etc. By extending the Landau dynamics approach [Phys. Rev. Lett. 93, 240403 (2004)PRLTAO0031-900710.1103/PhysRevLett.93.240403], we show that it is possible to capture the transverse dynamical modes (the "Kelvin modes") of the undulation of this "soliton filament" within the higher-dimensional space. These are the transverse stability or instability modes and are the ones potentially responsible for the breakup of the soliton into structures such as vortices, vortex rings, etc. We present the theory and case examples in 2D and 3D, corroborating the results by numerical stability and dynamical computations.

Solitons riding on solitons and the quantum Newton's cradle.

The reduced dynamics for dark and bright soliton chains in the one-dimensional nonlinear Schrödinger equation is used to study the behavior of collective compression waves corresponding to Toda lattice solitons. We coin the term hypersoliton to describe such solitary waves riding on a chain of solitons. It is observed that in the case of dark soliton chains, the formulated reduction dynamics provides an accurate an robust evolution of traveling hypersolitons. As an application to Bose-Einstein condensates trapped in a standard harmonic potential, we study the case of a finite dark soliton chain confined at the center of the trap. When the central chain is hit by a dark soliton, the energy is transferred through the chain as a hypersoliton that, in turn, ejects a dark soliton on the other end of the chain that, as it returns from its excursion up the trap, hits the central chain repeating the process. This periodic evolution is an analog of the classical Newton's cradle.

Proper Orthogonal Decomposition Methods for the Analysis of Real-Time Data: Exploring Peak Clustering in a Secondhand Smoke Exposure Intervention.

This work explores a method for classifying peaks appearing within a data-intensive time-series. We summarize a case study from a clinical trial aimed at reducing secondhand smoke exposure via the installation of air particle monitors in households. Proper orthogonal decomposition (POD) in conjunction with a k-means clustering algorithm assigns each data peak to one of two clusters. Aversive feedback from the monitors increased the proportion of short-duration, attenuated peaks from 38.8% to 96.6%. For each cluster, a distribution of parameters from a physics-based model of airborne particles is estimated. Peaks generated from these distributions are correctly identified by POD/clustering with >60% accuracy.

A tale of two distributions: from few to many vortices in quasi-two-dimensional Bose-Einstein condensates.

Motivated by the recent successes of particle models in capturing the precession and interactions of vortex structures in quasi-two-dimensional Bose-Einstein condensates, we revisit the relevant systems of ordinary differential equations. We consider the number of vortices N as a parameter and explore the prototypical configurations ('ground states') that arise in the case of few or many vortices. In the case of few vortices, we modify the classical result illustrating that vortex polygons in the form of a ring are unstable for N≥7. Additionally, we reconcile this modification with the recent identification of symmetry-breaking bifurcations for the cases of N=2,…,5. We also briefly discuss the case of a ring of vortices surrounding a central vortex (so-called N+1 configuration). We finally examine the opposite limit of large N and illustrate how a coarse-graining, continuum approach enables the accurate identification of the radial distribution of vortices in that limit.

From nodeless clouds and vortices to gray ring solitons and symmetry-broken states in two-dimensional polariton condensates.

We consider the existence, stability and dynamics of the nodeless state and fundamental nonlinear excitations, such as vortices, for a quasi-two-dimensional polariton condensate in the presence of pumping and nonlinear damping. We find a series of interesting features that can be directly contrasted to the case of the typically energy-conserving ultracold alkali-atom Bose-Einstein condensates (BECs). For sizeable parameter ranges, in line with earlier findings, the nodeless state becomes unstable towards the formation of stable nonlinear single or multi-vortex excitations. The potential instability of the single vortex is also examined and is found to possess similar characteristics to those of the nodeless cloud. We also report that, contrary to what is known, e.g., for the atomic BEC case, stable stationary gray ring solitons (that can be thought of as radial forms of Nozaki-Bekki holes) can be found for polariton condensates in suitable parametric regimes. In other regimes, however, these may also suffer symmetry-breaking instabilities. The dynamical, pattern-forming implications of the above instabilities are explored through direct numerical simulations and, in turn, give rise to waveforms with triangular or quadrupolar symmetry.

Directed ratchet transport in granular chains.

Directed-ratchet transport (DRT) in a one-dimensional lattice of spherical beads, which serves as a prototype for granular chains, is investigated. We consider a system where the trajectory of the central bead is prescribed by a biharmonic forcing function with broken time-reversal symmetry. By comparing the mean integrated force of beads equidistant from the forcing bead, two distinct types of directed transport can be observed-spatial and temporal DRT. Based on the value of the frequency of the forcing function relative to the cutoff frequency, the system can be categorized by the presence and magnitude of each type of DRT. Furthermore, we investigate and quantify how varying additional parameters such as the biharmonic weight affects DRT velocity and magnitude. Finally, friction is introduced into the system and is found to significantly inhibit spatial DRT. In fact, for sufficiently low forcing frequencies, the friction may even induce a switching of the DRT direction.

Exploring rigidly rotating vortex configurations and their bifurcations in atomic Bose-Einstein condensates.

In the present work, we consider the problem of a system of few vortices N ≤ 5 as it emerges from its experimental realization in the field of atomic Bose-Einstein condensates. Starting from the corresponding equations of motion for an axially symmetric trapped condensate, we use a two-pronged approach in order to reveal the configuration space of the system's preferred dynamical states. We use a Monte Carlo method parametrizing the vortex particles by means of hyperspherical coordinates and identifying the minimal energy ground states thereof for N=2,...,5 and different vortex particle angular momenta. We then complement this picture with a dynamical system analysis of the possible rigidly rotating states. The latter reveals a supercritical and subcritical pitchfork, as well as saddle-center bifurcations that arise, exposing the full wealth of the problem even for such low-dimensional cases. By corroborating the results of the two methods, it becomes fairly transparent which branch the Monte Carlo approach selects for different values of the angular momentum that is used as a bifurcation parameter.

Nonlinear localized modes in two-dimensional electrical lattices.

We report the observation of spontaneous localization of energy in two spatial dimensions in the context of nonlinear electrical lattices. Both stationary and moving self-localized modes were generated experimentally and theoretically in a family of two-dimensional square as well as honeycomb lattices composed of 6 × 6 elements. Specifically, we find regions in driver voltage and frequency where stationary discrete breathers, also known as intrinsic localized modes (ILMs), exist and are stable due to the interplay of damping and spatially homogeneous driving. By introducing additional capacitors into the unit cell, these lattices can controllably induce mobile discrete breathers. When more than one such ILMs are experimentally generated in the lattice, the interplay of nonlinearity, discreteness, and wave interactions generates a complex dynamics wherein the ILMs attempt to maintain a minimum distance between one another. Numerical simulations show good agreement with experimental results and confirm that these phenomena qualitatively carry over to larger lattice sizes.

Dynamics of a few corotating vortices in Bose-Einstein condensates.

We study the dynamics of small vortex clusters with a few (2-4) corotating vortices in Bose-Einstein condensates by means of experiments, numerical computations, and theoretical analysis. All of these approaches corroborate the counterintuitive presence of a dynamical instability of symmetric vortex configurations. The instability arises as a pitchfork bifurcation at sufficiently large values of the vortex system angular momentum that induces the emergence and stabilization of asymmetric rotating vortex configurations. The latter are quantified in the theoretical model and observed in the experiments. The dynamics is explored both for the integrable two-vortex particle system, where a reduction of the phase space of the system provides valuable insight, as well as for the nonintegrable three- (or more) vortex case, which additionally admits the possibility of chaotic trajectories.

Characteristics of two-dimensional quantum turbulence in a compressible superfluid.

Fluids subjected to suitable forcing will exhibit turbulence, with characteristics strongly affected by the fluid's physical properties and dimensionality. In this work, we explore two-dimensional (2D) quantum turbulence in an oblate Bose-Einstein condensate confined to an annular trapping potential. Experimentally, we find conditions for which small-scale stirring of the condensate generates disordered 2D vortex distributions that dissipatively evolve toward persistent currents, indicating energy transport from small to large length scales. Simulations of the experiment reveal spontaneous clustering of same-circulation vortices and an incompressible energy spectrum with k(-5/3) dependence for low wave numbers k. This work links experimentally observed vortex dynamics with signatures of 2D turbulence in a compressible superfluid.

Generation of localized modes in an electrical lattice using subharmonic driving.

We show experimentally and numerically that an intrinsic localized mode (ILM) can be stably produced (and experimentally observed) via subharmonic, spatially homogeneous driving in the context of a nonlinear electrical lattice. The precise nonlinear spatial response of the system has been seen to depend on the relative location in frequency between the driver frequency, ω(d), and the bottom of the linear dispersion curve, ω(0). If ω(d)/2 lies just below ω(0), then a single ILM can be generated in a 32-node lattice, whereas, when ω(d)/2 lies within the dispersion band, a spatially extended waveform resembling a train of ILMs results. To our knowledge, and despite its apparently broad relevance, such an experimental observation of subharmonically driven ILMs has not been previously reported.

Discrete breathers in a nonlinear electric line: modeling, computation, and experiment.

We study experimentally and numerically the existence and stability properties of discrete breathers in a periodic nonlinear electric line. The electric line is composed of single cell nodes, containing a varactor diode and an inductor, coupled together in a periodic ring configuration through inductors and driven uniformly by a harmonic external voltage source. A simple model for each cell is proposed by using a nonlinear form for the varactor characteristics through the current and capacitance dependence on the voltage. For an electrical line composed of 32 elements, we find the regions, in driver voltage and frequency, where n-peaked breather solutions exist and characterize their stability. The results are compared to experimental measurements with good quantitative agreement. We also examine the spontaneous formation of n-peaked breathers through modulational instability of the homogeneous steady state. The competition between different discrete breathers seeded by the modulational instability eventually leads to stationary n-peaked solutions whose precise locations is seen to sensitively depend on the initial conditions.

Dissipative solitary waves in granular crystals.

We provide a quantitative characterization of dissipative effects in one-dimensional granular crystals. We use the propagation of highly nonlinear solitary waves as a diagnostic tool and develop optimization schemes that allow one to compute the relevant exponents and prefactors of the dissipative terms in the equations of motion. We thereby propose a quantitatively accurate extension of the Hertzian model that encompasses dissipative effects via a discrete Laplacian of the velocities. Experiments and computations with steel, brass, and polytetrafluoroethylene reveal a common dissipation exponent with a material-dependent prefactor.

Controlling chaos of a Bose-Einstein condensate loaded into a moving optical Fourier-synthesized lattice.

We study the chaotic properties of steady-state traveling-wave solutions of the particle number density of a Bose-Einstein condensate with an attractive interatomic interaction loaded into a traveling optical lattice of variable shape. We demonstrate theoretically and numerically that chaotic traveling steady states can be reliably suppressed by small changes of the traveling optical lattice shape while keeping the remaining parameters constant. We find that the regularization route as the optical lattice shape is continuously varied is fairly rich, including crisis phenomena and period-doubling bifurcations. The conditions for a possible experimental realization of the control method are discussed.

Surface solitons in three dimensions.

We study localized modes on the surface of a three-dimensional dynamical lattice. The stability of these structures on the surface is investigated and compared to that in the bulk of the lattice. Typically, the surface makes the stability region larger, an extreme example of that being the three-site "horseshoe"-shaped structure, which is always unstable in the bulk, while at the surface it is stable near the anticontinuum limit. We also examine effects of the surface on lattice vortices. For the vortex placed parallel to the surface, the increased stability-region feature is also observed, while the vortex cannot exist in a state normal to the surface. More sophisticated localized dynamical structures, such as five-site horseshoes and pyramids, are also considered.

Solitons in one-dimensional nonlinear Schrödinger lattices with a local inhomogeneity.

In this paper we analyze the existence, stability, dynamical formation, and mobility properties of localized solutions in a one-dimensional system described by the discrete nonlinear Schrödinger equation with a linear point defect. We consider both attractive and repulsive defects in a focusing lattice. Among our main findings are (a) the destabilization of the on-site mode centered at the defect in the repulsive case, (b) the disappearance of localized modes in the vicinity of the defect due to saddle-node bifurcations for sufficiently strong defects of either type, (c) the decrease of the amplitude formation threshold for attractive and its increase for repulsive defects, and (d) the detailed elucidation as a function of initial speed and defect strength of the different regimes (trapping, trapping and reflection, pure reflection, and pure transmission) of interaction of a moving localized mode with the defect.

Discrete surface solitons in two dimensions.

We investigate fundamental localized modes in two-dimensional lattices with an edge (surface). The interaction with the edge expands the stability area for fundamental solitons, and induces a difference between dipoles oriented perpendicular and parallel to the surface. On the contrary, lattice vortex solitons cannot exist too close to the border. We also show, analytically and numerically, that the edge supports a species of localized patterns, which exists too but is unstable in the uniform lattice, namely, a horseshoe-shaped soliton, whose "skeleton" consists of three lattice sites. Unstable horseshoes transform themselves into a pair of ordinary solitons.