arXiv subjects

J. -G. Caputo

Publications and source records attributed to J. -G. Caputo.

At least 19 recordsLinked to original sources

arXiv2026

Blocking of 2D bistable reaction-diffusion fronts by obstacles

We investigate numerically the blocking of two-dimensional bistable reaction diffusion fronts by geometric obstacles. Our goal is to derive quantitative criteria for front propagation in the presence of spatial heterogeneities. Using a conservation law approach, we show that the integral of the reaction term acts as an effective driving force for the front. Combining this insight with the exact one-dimensional traveling wave solution, we construct a reduced analytical model that predicts blocking thresholds. In particular, we obtain explicit conditions for front propagation in a waveguide connected to a conical region of angle theta, valid for widths w less than 4. The model captures the influence of both geometry and nonlinearity, and shows good agreement with numerical simulations. Finally, we extend the analysis to more complex geometries, including checkerboard-like obstacles, and derive simple heuristic rules governing front propagation. ~

math-ph↗

arXiv2023

Localized eigenvectors on metric graphs

Using our previously published algorithm, we analyze the eigenvectors of the generalized Laplacian for two metric graphs occurring in practical applications. As expected, localization of an eigenvector is rare and the network should be tuned to observe exactly localized eigenvectors. We derive the resonance conditions to obtain localized eigenvectors for various geometric configurations and their combinations to form more complicated resonant structures. These localized eigenvectors suggest a new localization indicator based on the $L_2$ norm. They also can be excited, even with leaky boundary conditions, as shown by the numerical solution of the time-dependent wave equation on the metric graph. Finally, the study suggests practical ways to make resonating systems based on metric graphs.

math-ph↗

arXiv2023

Eigenvectors of graph Laplacians: a landscape

We review the properties of eigenvectors for the graph Laplacian matrix, aiming at predicting a specific eigenvalue/vector from the geometry of the graph. After considering classical graphs for which the spectrum is known, we focus on eigenvectors that have zero components and extend the pioneering results of Merris (1998) on graph transformations that preserve a given eigenvalue $λ$ or shift it in a simple way. These transformations enable us to obtain eigenvalues/vectors combinatorially instead of numerically; in particular we show that graphs having eigenvalues $λ= 1,2,\dots,6$ up to six vertices can be obtained from a short list of graphs. For the converse problem of a $λ$ subgraph $G$ of a $λ$ graph $G"$, we prove results and conjecture that $G$ and $G"$ are connected by two of the simple transformations described above.

math.SP↗

arXiv2021

Spectral solutions of PDEs on networks

To solve linear PDEs on metric graphs with standard coupling conditions (continuity and Kirchhoff's law), we develop and compare a spectral, a second-order finite difference, and a discontinuous Galerkin method. The spectral method yields eigenvalues and eigenvectors of arbitary order with machine precision and converges exponentially. These eigenvectors provide a Fourier-like basis on which to expand the solution; however, more complex coupling conditions require additional research. The discontinuous Galerkin method provides approximations of arbitrary polynomial order; however computing high-order eigenvalues accurately requires the respective eigenvector to be well-resolved. The method allows arbitrary non-Kirchhoff flux conditions and requires special penalty terms at the vertices to enforce continuity of the solutions. For the finite difference method, the standard one-sided second-order finite difference stencil reduces the accuracy of the vertex solution to $ O(h^{3/2})$. To preserve overall second-order accuracy, we used ghost cells for each edge. For all three methods we provide the implementation details, their validation, and examples illustrating their performance for the eigenproblem, Poisson equation, and the wave equation.

math.NA↗

arXiv2020

Breather stripes and radial breathers of the two-dimensional sine-Gordon equation

We revisit the problem of transverse instability of a 2D breather stripe of the sine-Gordon (sG) equation. A numerically computed Floquet spectrum of the stripe is compared to analytical predictions developed by means of multiple-scale perturbation theory showing good agreement in the long-wavelength limit. By means of direct simulations, it is found that the instability leads to a breakup of the quasi-1D breather in a chain of interacting 2D radial breathers that appear to be fairly robust in the dynamics. The stability and dynamics of radial breathers in a finite domain are studied in detail by means of numerical methods. Different families of such solutions are identified. They develop small-amplitude spatially oscillating tails ("nanoptera") through a resonance of higher-order breather's harmonics with linear modes ("phonons") belonging to the continuous spectrum. These results demonstrate the ability of the 2D sG model within our finite domain computations to localize energy in long-lived, self-trapped breathing excitations.

nlin.PS↗

arXiv2020

Epidemic model on a network: analysis and applications to COVID-19

We analyze an epidemic model on a network consisting of susceptible-infected-recovered equations at the nodes coupled by diffusion using a graph Laplacian. We introduce an epidemic criterion and examine different vaccination/containment strategies: we prove that it is most effective to vaccinate a node of highest degree. The model is also useful to evaluate deconfinement scenarios and prevent a so-called second wave. The model has few parameters enabling fitting to the data and the essential ingredient of importation of infected; these features are particularly important for the current COVID-19 epidemic.

physics.soc-ph↗

arXiv2020

Spectra of chains connected to complete graphs

We characterize the spectrum of the Laplacian of graphs composed of one or two finite or infinite chains connected to a complete graph. We show the existence of localized eigenvectors of two types, eigenvectors that vanish exactly outside the complete graph and eigenvectors that decrease exponentially outside the complete graph. Our results also imply gaps between the eigenvalues corresponding to localized and extended eigenvectors.

math.SP↗

arXiv2018

Planar and Radial Kinks in Nonlinear Klein-Gordon Models: Existence, Stability and Dynamics

We consider effectively one-dimensional planar and radial kinks in two-dimensional nonlinear Klein-Gordon models and focus on the sine-Gordon model and the $ϕ^4$ variants thereof. We adapt an adiabatic invariant formulation recently developed for nonlinear Schr{ö}dinger equations, and we study the transverse stability of these kinks. This enables us to characterize one-dimensional planar kinks as solitonic filaments, whose stationary states and corresponding spectral stability can be characterized not only in the homogeneous case, but also in the presence of external potentials. Beyond that, the full nonlinear (transverse) dynamics of such filaments are described using the reduced, one-dimensional, adiabatic invariant formulation. For radial kinks, this approach confirms their azimuthal stability. It also predicts the possibility of creating stationary and stable ring-like kinks. In all cases we corroborate the results of our methodology with full numerics on the original sine-Gordon and $ϕ^4$ models.

nlin.PS↗

arXiv2015

Polarization rotation by an rf-SQUID metasurface

We study the transmission and reflection of a plane electromagnetic wave through a two dimensional array of rf-SQUIDs. The basic equations describing the amplitudes of the magnetic field and current in the split-ring resonators are developed. These yield in the linear approximation the reflection and transmission coefficients. The polarization of the reflected wave is independent of the frequency of the incident wave and of its polarization; it is defined only by the orientation of the split-ring. The reflection and transmission coefficients have a strong resonance that is determined by the parameters of the rf-SQUID; its strength depends essentially on the incident angle.

physics.optics↗

arXiv2014

Bistable reaction-diffusion on a network

We study analytically and numerically a bistable reaction-diffusion equation on an arbitrary finite network. We prove that stable fixed points (multi-fronts) exist for any configuration as long as the diffusion is small. We also study fold bifurcations leading to depinning and give a simple depinning criterion. These results are confirmed by using continuation techniques from bifurcation theory and by solving the time dependent problem near the treshold. A qualitative comparison principle is proved and verified for time dependent solutions, and for some related models.

nlin.AO↗

arXiv2013

Screening magnetic fields by a superconducting disk: a simple model

We introduce a simple approach to evaluate the magnetic field distribution around superconducting samples, based on the London equations; the elementary variable is the vector potential. This procedure has no adjustable parameters, only the sample geometry and the London length, $λ$, determine the solution. The calculated field reproduces quantitatively the measured induction field above MgB$_2$ disks of different diameters, at 20K and for applied fields lower than 0.4T. The model can be applied if the flux line penetration inside the sample can be neglected when calculating the induction field distribution outside the superconductor. Finally we show on a cup-shape geometry how one can design a magnetic shield satisfying a specific constraint.

cond-mat.supr-con↗

arXiv2013

Radial sine-Gordon kinks as sources of fast breathers

We consider radial sine-Gordon kinks in two, three and higher dimensions. A full two dimensional simulation showing that azimuthal perturbations remain small allows to reduce the problem to the one dimensional radial sine-Gordon equation. We solve this equation on an interval $[r_0,r_1]$ and absorb all outgoing radiation. Before collision the kink is well described by a simple law derived from the conservation of energy. In two dimensions for $r_0 \le 2$, the collision disintegrates the kink into a fast breather while for $r_0 \ge 4$ we obtain a kink-breather meta-stable state where breathers are shed at each kink "return". In three and higher dimensions $d$ a kink-pulson state appears for small $r_0$. The three states then exist as shown by a study of the $(d,r_0)$ parameter space. On the application side, the kink disintegration opens the way for new types of terahertz microwave generators.

nlin.PS↗

arXiv2012

Fast control of the reflection of a ferroelectric by an extremely short pulse

We propose a new type of optical switch based on a ferroelectric. It is based on the gap which exists for waves propagating from a dielectric to a ferroelectric material. This gap depends on the polarization of the ferroelectric. We show that it can be shifted by a control electromagnetic pulse so that the material becomes transparent. This device would shift much faster than the relaxation time of the ferroelectric (1 nano s). Estimates are given for a real material.

physics.optics↗

arXiv2012

Electrodynamics of a split-ring Josephson resonator in a microwave line

We consider the coupling of an electromagnetic wave to a split-ring Josephson oscillator or radio-frequency SQUID in the hysteretic regime. This device is similar to an atomic system in that it has a number of steady states. We show that one can switch between these with a suitable short external microwave pulse. The steady states can be characterized by their resonant lines which are of the Fano type. Using a static magnetic field, we can shift these spectral lines and lift their degeneracy.

cond-mat.mes-hall↗

arXiv2010

High frequency polarization switching of a thin ferroelectric film

We consider both experimentally and analytically the transient oscillatory process that arises when a rapid change in voltage is applied to a $Ba_xSr_{1-x}TiO_3$ ferroelectric thin film deposited on an $Mg0$ substrate. High frequency ($\approx 10^{8} rad/s$) polarization oscillations are observed in the ferroelectric sample. These can be understood using a simple field-polarization model. In particular we obtain analytic expressions for the oscillation frequency and the decay time of the polarization fluctuation in terms of the material parameters. These estimations agree well with the experimental results.

cond-mat.mtrl-sci↗

arXiv2010

Dynamics of point Josephson junctions in a microstrip line

We analyze a new long wave model describing the electrodynamics of an array of point Josephson junctions in a superconducting cavity. It consists in a wave equation with Dirac delta function sine nonlinearities. We introduce an adapted spectral problem whose spectrum gives the resonances in the current-voltage characteristic curve of any array. Using the associated inner product and eigenmodes, we establish that at the resonances the solution is described by two simple ordinary differential equations.

cond-mat.supr-con↗

arXiv2009

Fast electromagnetic response of a thin film of resonant atoms with permanent dipole

We consider the propagation of extremely short pulses through a dielectric thin film containing resonant atoms (two level atoms) with permanent dipole. Assuming that the film width is less than the field wave length, we can solve the wave equation and reduce the problem to a system of generalized Bloch equations describing the resonant atoms. We compute the stationary solutions for a constant irradiation of the film. Superimposing a small amplitude linear wave we compute the reflection and transmission coefficients. From these, one can then deduce the different parameters of the model. We believe this technique could be used in experiments to obtain the medium atomic and relaxation parameters.

nlin.PS↗

arXiv2008

Cavity with an embedded polarized film: an adapted spectral approach

We consider the modes of the electric field of a cavity where there is an embedded polarized dielectric film. The model consists in the Maxwell equations coupled to a Duffing oscillator for the film which we assume infinitely thin. We derive the normal modes of the system and show that they are orthogonal with a special scalar product which we introduce. These modes are well suited to describe the system even for a film of finite thickness. By acting on the film we demonstrate switching from one cavity mode to another. Since the system is linear, little energy is needed for this conversion. Moreover the amplitude equations describe very well this complex system under different perturbations (damping, forcing and nonlinearity) with very few modes. These results are very general and can be applied to different situations like for an atom in a cavity or a Josephson junction in a capacitor and this could be very useful for many nano-physics applications.

physics.optics↗