arXiv subjects

Gregory Kozyreff

Publications and source records attributed to Gregory Kozyreff.

12 recordsLinked to original sources

arXiv2026

On the independence of the slow and fast scales in multiple-scale expansions, with application to Van der Pol's equation

When implementing the method of multiple scales, one is traditionally instructed to treat the slow and fast time scales as if they were independent. Despite the intuitive motivation and the effectiveness of this perturbation method, one cannot failt to notice that these two scales relate to the same unique variable, so independence can only be formal. How sensible is it, then, to split a variable asymptotically into two (or more) independent ones? In this paper, we elucidate this issue with Van der Pol's equation, one of the simplest weakly nonlinear oscillators, as well as a simple example of a Hopf bifurcation. The discussion involves carrying the multiple-scale analysis up to arbitrarily large order and dealing with the divergent character of the resulting asymptotic series. Using the technique of optimal truncation, we re-connect the two scales. Specifically, we show that an initial translation of the fast coordinate leads to a non-trivial, exponentially small, phase shift that depends on the slow coordinate. This phase shift breaks the independence of the slow and fast scales and is found to result from the nonlinearity. Numerical simulations confirm its existence, as well as the predicted scaling. The calculation is carried out in sufficient detail to provide confidence in the generality of our result, both in its essence and in its form. In particular, we find strong indications that a Hopf bifurcation with a quadratic nonlinearity would lead to the same phenomenon, but with a larger magnitude.

math.DS↗

arXiv2023

Multiple-scale analysis of the simplest large-delay differential equation

A delayed term in a differential equation reflects the fact that information takes significant time to travel from one place to another within a process being studied. Despite de apparent similarity with ordinary differential equations, delay-differential equations (DDE) are known to be fundamentally different and to require a dedicate mathematical apparatus for their analysis. Indeed, when the delay is large, it was found that they can sometimes be related to spatially extended dynamical systems. The purpose of this paper is to explain this fact in the simplest possible DDE by way of a multiple-scale analysis. We show the asymptotic correspondence of that linear DDE with the diffusion equation. This partial differential equations arises from a solvability condition that differs from the ones usually encountered in textbooks on asymptotics: In the limit of large delays, the leading-order problem is a map and secular divergence at subsequent orders stem from forcing terms in that map.

math.DS↗

arXiv2023

The θ-formulation of the 2D elastica -- Buckling and boundary layer theory

The equations of a planar elastica under pressure can be rewritten in a useful form by parametrising the variables in terms of the local orientation angle, $θ$, instead of the arc length. This ``$θ$-formulation'' lends itself to a particularly easy boundary layer analysis in the limit of weak bending stiffness. Within this parameterization, boundary layers are located at inflexion points, where $θ$ is extremum, and they connect regions of low and large curvature. A simple composite solution is derived without resorting to elliptic functions and integrals. This approximation can be used as an elementary building block to describe complex shapes. Applying this theory to the study of an elastic ring under uniform pressure and subject to a set of point forces, we discover a snapping instability. This instability is confirmed by numerical simulations. Finally, we carry out experiments and find good agreement of the theory with the experimental shape of the deformed elastica.

cond-mat.soft↗

arXiv2022

On the speed of wave packets and the nonlinear Schrödinger equation

The universal theory of weakly nonlinear wave packets given by the nonlinear Schrödinger equation is revisited. In the limit where the group and phase velocities are very close together, a multiple scale analysis carried out beyond all orders reveals that a single soliton, bright or dark, can travel at a different speed than the group velocity. In an exponentially small but finite range of parameters, the envelope of the soliton is locked to the rapid oscillations of the carrier wave. Eventually, the dynamics is governed by an equation anologous to that of a pendulum, in which the center of mass of the soliton is subjected to a periodic potential. Consequently, the soliton speed is not constant and generally contains a periodic component. Furthermore, the interaction between two distant solitons can in principle be profoundly altered by the aforementioned effective periodic potential and we conjecture the existence of new bound states. These results are derived on a wide class of wave models and in such a general way that they are believed to be of universal validity.

nlin.PS↗

arXiv2022

Effect of external tension on the wetting of an elastic sheet

Recent studies of elasto-capillary phenomena have triggered interest in a basic variant of the classical Young-Laplace-Dupré (YLD) problem: The capillary interaction between a liquid drop and a thin solid sheet of low bending stiffness. Here, we consider a two-dimensional model where the sheet is subjected to an external tensile load and the drop is characterized by a well-defined Young's contact angle $θ_Y$. Using a combination of numerical, variational, and asymptotic techniques, we discuss wetting as a function of the applied tension. We find that, for wettable surfaces with $0<θ_Y<π/2$, complete wetting is possible below a critical applied tension thanks to the deformation of the sheet in contrast with rigid substrates requiring $θ_Y=0$. Conversely, for very large applied tensions, the sheet becomes flat and the classical YLD situation of partial wetting is recovered. At intermediate tensions, a vesicle forms in the sheet, which encloses most of the fluid and we provide an accurate asymptotic description of this wetting state in the limit of small bending stiffness. We show that bending stiffness, however small, affects the entire shape of the vesicle. Rich bifurcation diagrams involving partial wetting and ``vesicle'' solution are found. For moderately small bending stiffnesses, partial wetting can coexist both with the vesicle solution and complete wetting. Finally, we identify a tension-dependent bendo-capillary length, $λ_\text{BC}$, and find that the shape of the drop is determined by the ratio $A/λ_\text{BC}^2$, where $A$ is the area of the drop.

cond-mat.soft↗

arXiv2021

Dynamical elastic contact of a rope with the ground

A rope laid on the ground with one end subjected to time-dependent forcing is proposed as a prototypical elastic dynamical contact problem, which we study analytically, numerically, and experimentally. The dynamics is governed by an infinite set of linear and nonlinear resonances. In the limit of weak bending stiffness, the fundamental frequency is found to be independent of the rope tension. A transition between a radiation-less and a wave radiating state occurs via a series of grazing bifurcations, whereby new contacts between the rope and the ground are formed. The grazing bifurcations form overlapping Arnold tongues in the frequency-amplitude parameter space. Finally, for ropes with large bending stiffness and when the geometric nonlinearity is important, bistability is observed between several wave-making regimes.

cond-mat.soft↗

arXiv2021

Hospitalization dynamics during the first COVID-19 pandemic wave: SIR modelling compared to Belgium, France, Italy, Switzerland and New York City data

Using the classical Susceptible-Infected-Recovered epidemiological model, an analytical formula is derived for the number of beds occupied by Covid-19 patients. The analytical curve is fitted to data in Belgium, France, New York City and Switzerland, with a correlation coefficient exceeding 98.8%, suggesting that finer models are unnecessary with such macroscopic data. The fitting is used to extract estimates of the doubling time in the ascending phase of the epidemic, the mean recovery time and, for those who require medical intervention, the mean hospitalization time. Large variations can be observed among different outbreaks.

q-bio.PE↗

arXiv2020

Berry Phases in the Reconstructed KdV Equation

We consider the KdV equation on a circle and its Lie-Poisson reconstruction, which is reminiscent of an equation of motion for fluid particles. For periodic waves, the stroboscopic reconstructed motion is governed by an iterated map whose Poincaré rotation number yields the drift velocity. We show that this number has a geometric origin: it is the sum of a dynamical phase, a Berry phase, and an "anomalous phase". The last two quantities are universal: they are solely due to the underlying Virasoro group structure. The Berry phase, in particular, was previously described in [arXiv:1703.06142] for two-dimensional conformal field theories, and follows from adiabatic deformations produced by the propagating wave. We illustrate these general results with cnoidal waves, for which all phases can be evaluated in closed form thanks to a uniformizing map that we derive. Along the way, we encounter "orbital bifurcations" occurring when a wave becomes non-uniformizable: there exists a resonance wedge, in the cnoidal parameter space, where particle motion is locked to the wave, while no such locking occurs outside of the wedge.

math-ph↗

arXiv2019

Large Q factor with very small Whispering Gallery Modes resonators

Efficient micro-resonators simultaneously require a large quality factor $Q$ and a small volume $V$. However, the former is ultimately limited by bending losses, the unavoidable radiation of energy of a wave upon changing direction of propagation. Such bending losses increase exponentially as $V$ decreases and eventually result in a drop of $Q$. Therefore, circular cavities are generally designed with radii that are much larger than the optical wavelength. The same leakage of energy by radiation limits the sharpness of bends in photonic integrated circuits. In this article, we present a way to reduce bending losses in circular micro-resonators. The proposed scheme consists of one or more external dielectric rings that are concentric with the cavity. These rings alter the field outside the cavity where radial oscillations set in, and thus control the far field radiation. As a result, the $Q$ factor can be increased by several orders of magnitude while keeping a small cavity volume.

physics.optics↗

arXiv2017

Multiple critical coupling and sensing in a microresonator-waveguide system

We study the optical transmission of a waveguide that is side-coupled to a high-$Q$ circular microresonator. The coupling is critical if the intrinsic resonator losses equate the coupling losses to the waveguide. When this happens, the transmittance of the waveguide displays resonance dips with maximal depth as the frequency is swept through the resonators resonances. We show that multiple configurations, parameterised by the minimal distance between the resonator and the waveguide, can lead to critical coupling. Indeed, for a sufficiently large resonator radius, the flow of power between the waveguide and the resonator can change sign several times within a single pass. This leads to an oscillatory coupling parameter as a function of the separation distance. As a result, multiple geometrical configurations can lead to critical coupling, even if the waveguide lies in the equatorial plane of the resonator. These results are explained using coupled-mode theory and full wave numerical simulations. In the vicinity of secondary or higher-order critical coupling, the depth of the transmittance dip is very sensitive to the environment. We discuss how this effect can be exploited for sensing purpose. Alternatively, by actively controlling the environment in the secondary critical configuration, the waveguide/resonator system can be driven as an optical switch.

physics.optics↗

arXiv2017

Organic solar cell design as a function of radiative quantum efficiency

We study the radiative decay, or fluorescence, of excitons in organic solar cells as a function of its geometrical parameters. Contrary to their non-radiative counterpart, fluorescence losses strongly depend on the environment. By properly tuning the thicknesses of the buffer layers between the active regions of the cell and the electrodes, the exciton lifetime and, hence, the exciton diffusion length can be increased. The importance of this phenomenon depends on the radiative quantum efficiency, which is the fraction of the exciton decay that is intrinsically due to fluorescence. Besides this effect, interferences within the cell control the efficiency of sunlight injection into the active layers. An optimal cell design must rely on the consideration of these two aspects. By properly managing fluorescence losses, one can significantly improve the cell performance. To demonstrate this fact, we use realistic material parameters inspired from literature data and obtain an increase of power conversion efficiency from 11.3% to 12.7%. Conversely, not to take into account the strong dependence of fluorescence on the environment may lead to a sub-optimal cell design and a degradation of cell performance. The presence of radiative losses, however small, significantly changes the optimal thicknesses. We illustrate this latter situation with experimental material data.

physics.app-ph↗

arXiv2016

Dispersion relations and bending losses of cylindrical and spherical shells, slabs, and slot waveguides

We derive formulas for Whispering Gallery Mode resonances and bending losses in infinite cylindrical dielectric shells and sets of concentric cylindrical shells. The formulas also apply to spherical shells and to sections of bent waveguides. The derivation is based on a WKB treatment of Helmholtz equation and can in principle be extended to any number of concentric shells. A distinctive limit analytically arises in the analysis when two shells are brought at very close distance to one another. In that limit, the two shells act as a slot waveguide. If the two shells are sufficiently apart, we identify a structural resonance between the individual shells, which can either lead to a substantial enhancement or suppression of radiation losses.

physics.optics↗