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Abstract We develop a mathematical model of heat and mass transfer in a configuration which involves a spherical droplet levitating near a flat liquid layer heated from below. Analytical solutions for vapor concentration in air and the temperature distributions both inside the droplet and in moist air around it are coupled to the numerical solution for heat transfer in the liquid layer. In the limit of weak evaporation, the liquid layer surface is cooled locally due to the presence of the droplet, while the effect is reversed for strong evaporation. The latter case is also characterized by higher temperature near the bottom of the droplet and stronger temperature gradients in the droplet itself, an unexpected conclusion given the high liquid-to-air thermal conductivity ratio. The observations are explained in terms of interplay between geometric and thermal effects of the presence of the droplet. A simple analytical criterion is formulated to determine the condition when the droplet presence has no effect on the layer temperature; remarkably, the condition does not depend on the distance between the droplet and the layer surface. The calculation of the evaporation rate leads to determination of the flow around the droplet, treated in the Stokes flow approximation, as well as the levitation height. 

Abstract We prove that the solutions to the discrete nonlinear Schrödinger equation with non-local algebraically decaying coupling converge strongly in$$L^2({\mathbb {R}}^2)$$ $L^2\left(R^2\right)$to those of the continuum fractional nonlinear Schrödinger equation, as the discretization parameter tends to zero. The proof relies on sharp dispersive estimates that yield the Strichartz estimates that are uniform in the discretization parameter. An explicit computation of the leading term of the oscillatory integral asymptotics is used to show that the best constants of a family of dispersive estimates blow up as the non-locality parameter$$\alpha \in (1,2)$$ $\alpha \in (1,2)$approaches the boundaries. 

Abstract We consider the existence and spectral stability of nonlinear discrete localized solutions representing light pulses propagating in a twisted multicore optical fiber. By considering an even number,N, of waveguides, we derive asymptotic expressions for solutions in which the bulk of the light intensity is concentrated as soliton‐like pulses confined to a single waveguide. The leading order terms obtained are in very good agreement with results of numerical computations. Furthermore, as in the model without temporal dispersion, when the twist parameter, ϕ, is given by , these standing waves exhibit optical suppression, in which a single waveguide remains unexcited, to leading order. Spectral computations and numerical evolution experiments suggest that these standing wave solutions are stable for values of the coupling parameter less than a critical value, at which point a spectral instability results from the collision of an internal eigenvalue with the eigenvalues at the origin. This critical value has a maximum when . 

Topological photonics is a framework that follows both condensed matter physics and topology. It refers to designing the guiding properties of the propagating medium (e.g., a photonic crystal or a waveguide lattice) in such a way that the transport of electromagnetic energy is realized in unique, robust, and sometimes unexpected ways. Consider a simple thought experiment: imagine first the two-dimensional wave equation on a square domain, and assume homogeneous Dirichlet boundary conditions. We know that the accessible modes extend in periodic form throughout the whole domain and, in time, waves can propagate in all directions. This behavior is in response to the inherent symmetries of the medium. Imagine instead that we engineer the medium in such a way that all the energy concentrates in the boundary of the medium and propagates in only one direction. (In the language of optics, this would be seen as inhibiting back reflection and making the bulk medium act like an insulator). 

In the work of Colliander et al. (2020) a minimal lattice model was constructed describing the transfer of energy to high frequencies in the defocusing nonlinear Schrödinger equation. In the present work, we present a systematic study of the coherent structures, both standing and traveling, that arise in the context of this model. We find that the nonlinearly dispersive nature of the model is responsible for standing waves in the form of discrete compactons. On the other hand, analysis of the dynamical features of the simplest nontrivial variant of the model, namely the dimer case, yields both solutions where the intensity is trapped in a single site and solutions where the intensity moves between the two sites, which suggests the possibility of moving excitations in larger lattices. Such excitations are also suggested by the dynamical evolution associated with modulational instability. Our numerical computations confirm this expectation, and we systematically construct such traveling states as exact solutions in lattices of varying size, as well as explore their stability. A remarkable feature of these traveling lattice waves is that they are of ‘‘antidark’’ type, i.e., they are mounted on top of a non-vanishing background. These studies shed light on the existence, stability and dynamics of such standing and traveling states in 1 + 1 dimensions, and pave the way for exploration of corresponding configurations in higher dimensions. 

In the work of Colliander et al. (2020) a minimal lattice model was constructed describing the transfer of energy to high frequencies in the defocusing nonlinear Schrödinger equation. In the present work, we present a systematic study of the coherent structures, both standing and traveling, that arise in the context of this model. We find that the nonlinearly dispersive nature of the model is responsible for standing waves in the form of discrete compactons. On the other hand, analysis of the dynamical features of the simplest nontrivial variant of the model, namely the dimer case, yields both solutions where the intensity is trapped in a single site and solutions where the intensity moves between the two sites, which suggests the possibility of moving excitations in larger lattices. Such excitations are also suggested by the dynamical evolution associated with modulational instability. Our numerical computations confirm this expectation, and we systematically construct such traveling states as exact solutions in lattices of varying size, as well as explore their stability. A remarkable feature of these traveling lattice waves is that they are of ‘‘antidark’’ type, i.e., they are mounted on top of a non-vanishing background. These studies shed light on the existence, stability and dynamics of such standing and traveling states in 1 + 1 dimensions, and pave the way for exploration of corresponding configurations in higher dimensions. 

In the present work, we study coherent structures in a one-dimensional discrete nonlinear Schrodinger lattice in which the coupling between waveguides is periodically modulated. Numerical experiments with single-site initial conditions show that, depending on the power, the system exhibits two fundamentally different behaviors. At low power, initial conditions with intensity concentrated in a single site give rise to transport, with the energy moving unidirectionally along the lattice, whereas high power initial conditions yield stationary solutions. We explain these two behaviors, as well as the nature of the transition between the two regimes, by analyzing a simpler model where the couplings between waveguides are given by step functions. For the original model, we numerically construct both stationary and moving coherent structures, which are solutions reproducing themselves exactly after an integer multiple of the coupling period. For the stationary solutions, which are true periodic orbits, we use Floquet analysis to determine the parameter regime for which they are spectrally stable. Typically, the traveling solutions are characterized by having small-amplitude, oscillatory tails, although we identify a set of parameters for which these tails disappear. These parameters turn out to be independent of the lattice size, and our simulations suggest that for these parameters, numerically exact traveling solutions are stable. 

In bistable perception, observers experience alternations between two interpretations of an unchanging stimulus. Neurophysiological studies of bistable perception typically partition neural measurements into stimulus-based epochs and assess neuronal differences between epochs based on subjects' perceptual reports. Computational studies replicate statistical properties of percept durations with modeling principles like competitive attractors or Bayesian inference. However, bridging neuro-behavioral findings with modeling theory requires the analysis of single-trial dynamic data. Here, we propose an algorithm for extracting nonstationary timeseries features from single-trial electrocorticography (ECoG) data. We applied the proposed algorithm to 5-min ECoG recordings from human primary auditory cortex obtained during perceptual alternations in an auditory triplet streaming task (six subjects: four male, two female). We report two ensembles of emergent neuronal features in all trial blocks. One ensemble consists of periodic functions that encode a stereotypical response to the stimulus. The other comprises more transient features and encodes dynamics associated with bistable perception at multiple time scales: minutes (within-trial alternations), seconds (duration of individual percepts), and milliseconds (switches between percepts). Within the second ensemble, we identified a slowly drifting rhythm that correlates with the perceptual states and several oscillators with phase shifts near perceptual switches. Projections of single-trial ECoG data onto these features establish low-dimensional attractor-like geometric structures invariant across subjects and stimulus types. These findings provide supporting neural evidence for computational models with oscillatory-driven attractor-based principles. The feature extraction techniques described here generalize across recording modality and are appropriate when hypothesized low-dimensional dynamics characterize an underlying neural system. SIGNIFICANCE STATEMENTIrrespective of the sensory modality, neurophysiological studies of multistable perception have typically investigated events time-locked to the perceptual switching rather than the time course of the perceptual states per se. Here, we propose an algorithm that extracts neuronal features of bistable auditory perception from largescale single-trial data while remaining agnostic to the subject's perceptual reports. The algorithm captures the dynamics of perception at multiple timescales, minutes (within-trial alternations), seconds (durations of individual percepts), and milliseconds (timing of switches), and distinguishes attributes of neural encoding of the stimulus from those encoding the perceptual states. Finally, our analysis identifies a set of latent variables that exhibit alternating dynamics along a low-dimensional manifold, similar to trajectories in attractor-based models for perceptual bistability.