arXiv subjects

Jean-Guy Caputo

Publications and source records attributed to Jean-Guy Caputo.

At least 19 recordsLinked to original sources

arXiv2026

Collective coordinate descriptions of a kink in a driven-damped $\phi^4$ model

Extending a recent effective theory formulation for the dynamics of kinks in the sine-Gordon model [1], we propose an analogous effective description of $\phi^4$ kinks. Three different reduced models based on the kink position, width and internal mode amplitude are introduced and compared systematically with the numerical solution of the equation with space- and time-dependent perturbations. In all cases considered, the model based on the kink position and width agrees the best with the full numerical solution. As long as the external driving frequency of the perturbation remains moderate, it captures with remarkable accuracy the intricate dynamical processes taking place in the system.

nlin.PS↗

arXiv2025

Numerical analysis of scattering on combinatorial graphs

We investigate numerically the scattering of waves on discrete graphs. An efficient algorithm is developed to compute the reflection and transmission (spectral) coefficients. We then explore various configurations of input and output leads, demonstrating how bound states arising from specific vertex-lead connections result in total reflection. The impedance of the leads is shown to influence the spectral coefficients in a predictable manner. Furthermore, for a given input lead we show that the total transmission of a wave packet can be maximized by appropriately selecting the exit lead. Finally, we analyze the spectral signatures of defects within the graph and find that they vary depending on both the defect's location and its spectral characteristics thus enabling its identification.

math-ph↗

arXiv2025

On some classes of bivalent and trivalent planar graphs

A graph is called bivalent or trivalent if there exists an eigenvector of the graph Laplacian composed from {-1,1} or {-1,0,1}, respectively. These bivalent and trivalent eigenvectors are important for engineering applications, in particular for vibrating systems. In this article, we determine the structure of bivalent and trivalent graphs in the following planar graph families: trees, unicyclic, bicyclic, and cactus.

math.CO↗

arXiv2024

Mathematical analysis of a flux-jump model in superconductivity

Type II superconductors can trap a transient magnetic field and become "cryomagnets" that are very useful for applications. During this process, flux jumps i.e. sudden jumps of the total magnetization occur and hinder the properties of these magnets. To understand the electrodynamics of these systems and in particular flux jumps, we analyzed mathematically a model based on Maxwell's equations and temperature in a 1D configuration. When a magnetic pulse is applied to a superconductor, three effects occur, from fastest to slowest: Joule heating, magnetic relaxation and temperature diffusion. Adimensionalising the problem, we obtain a nonlinear diffusion for the magnetic field coupled to a forced diffusion equation for the temperature with only two parameters. Two regimes occur, depending on temperature: for medium temperature the heat capacity of a sample can be assumed constant while for low temperature it depends on temperature causing a nonlinear temperature evolution. Flux jumps can be explained using the fixed points of the equations. We found that they occur for pulses of duration close to the magnetic relaxation time and mostly at low temperature because of the nonlinear dependance. Flux trapping is maximal for medium amplitude long duration pulses and low to medium temperatures, so these conditions are optimal to produce better cryomagnets.

cond-mat.supr-con↗

arXiv2024

Tracking disturbances in transmission networks

We study the nonlinear inverse source problem of detecting, localizing and identifying unknown accidental disturbances on forced and damped transmission networks. A first result is that strategic observation sets are enough to guarantee detection of disturbances. To localize and identify them, we additionally need the observation set to be absorbent. If this set is dominantly absorbent, then detection, localization and identification can be done in "quasi real-time". We illustrate these results with numerical experiments.

math.OC↗

arXiv2024

Radial kinks in a Schwarzschild-like geometry

We study the propagation of a domain wall (kink) of the $\phi^4$ model in a radially symmetric environment defined by a gravity source. This source deforms the standard Euclidian metric into a Schwarzschild-like one. We introduce an effective model that accurately describes the dynamics of the kink center. This description works well even outside the perturbation region, i.e., even for large masses of the gravitating object. We observed that such a spherical domain wall surrounding a star-type object inevitably "collapses", i.e., shrinks in radius towards the origin and offer an understanding of the latter phenomenology. The relevant analysis is presented for a circular domain wall and a spherical one.

nlin.PS↗

arXiv2024

Influence of gauges in the numerical simulation of the time-dependent Ginzburg-Landau model

The time-dependent Ginzburg-Landau (TDGL) model requires the choice of a gauge for the problem to be mathematically well-posed. In the literature, three gauges are commonly used: the Coulomb gauge, the Lorenz gauge and the temporal gauge. It has been noticed [J. Fleckinger-Pellé et al., Technical report, Argonne National Lab. (1997)] that these gauges can be continuously related by a single parameter considering the more general $ω$-gauge, where $ω$ is a non-negative real parameter. In this article, we study the influence of the gauge parameter $ω$ on the convergence of numerical simulations of the TDGL model using finite element schemes. A classical benchmark is first analysed for different values of $ω$ and artefacts are observed for lower values of $ω$. Then, we relate these observations with a systematic study of convergence orders in the unified $ω$-gauge framework. In particular, we show the existence of a tipping point value for $ω$, separating optimal convergence behaviour and a degenerate one. We find that numerical artefacts are correlated to the degeneracy of the convergence order of the method and we suggest strategies to avoid such undesirable effects. New 3D configurations are also investigated (the sphere with or without geometrical defect).

math.NA↗

arXiv2024

Mitigating an epidemic on a geographic network using vaccination

We consider a mathematical model describing the propagation of an epidemic on a geographical network. The size of the outbreak is governed by the initial growth rate of the disease given by the maximal eigenvalue of the epidemic matrix formed by the susceptibles and the graph Laplacian representing the mobility. We use matrix perturbation theory to analyze the epidemic matrix and define a vaccination strategy, assuming the vaccination reduces the susceptibles. When mobility and local disease dynamics have similar time scales, it is most efficient to vaccinate the whole network because the disease grows uniformly. However, if only a few vertices can be vaccinated then which ones do we choose? We answer this question, and show that it is most efficient to vaccinate along an eigenvector corresponding to the largest eigenvalue of the Laplacian. We illustrate these general results on a 7 vertex graph and a realistic example of the french rail network. When mobility is slower than local disease dynamics, the epidemic grows on the vertex with largest susceptibles. The epidemic growth rate is more reduced when vaccinating a larger degree vertex; it also depends on the neighboring vertices. This study and its conclusions provides guidelines for the planning of vaccination on a network at the onset of an epidemic.

physics.soc-ph↗

arXiv2021

An Abelian Higgs model of pulsed field magnetization in superconductors

Pulsed field magnetization leads to trapped magnetic field persistent for long times.We present a one-dimensional model of the interaction between an electromagnetic wave and a superconducting slab based on the Maxwell-Ginzburg-Landau (Abelian Higgs) theory. We first derive the model starting from a Lagrangian coupling the electromagnetic field with the Ginzburg-Landau potential for the superconductor. Then we explore numerically its capabilities by applying a Gaussian vector potential pulse and monitoring usual quantities such as the modulus and the phase of the order parameter. We also introduce defects in the computational domain. We show that the presence of defects enhances the remanent vector potential and diminishes the modulus of the order parameter, in agreement with existing experiments.

cond-mat.supr-con↗

arXiv2021

Analysis of trophic networks: an optimisation approach

We introduce a methodology to study the possible matter flows of an ecosystem defined by observational biomass data and realistic biological constraints. The flows belong to a polyhedron in a multi dimensional space making statistical exploration difficult in practice; instead, we propose to solve a convex optimization problem. Five criteria corresponding to ecological network indices have been selected to be used as convex goal functions. Numerical results show that the method is fast and can be used for large systems. Minimum flow solutions are analyzed using flow decomposition in paths and circuits. Their consistency is also tested by introducing a system of differential equations for the biomasses and examining the stability of the biomass fixed point. The method is illustrated and explained throughout the text on an ecosystem toy model. It is also applied to realistic food models.

nlin.AO↗

arXiv2020

Stopping a reaction-diffusion front

We revisit the problem of pinning a reaction-diffusion front by a defect, in particular by a reaction-free region. Using collective variables for the front and numerical simulations, we compare the behaviors of a bistable and monostable front. A bistable front can be pinned as confirmed by a pinning criterion, the analysis of the time independant problem and simulations. Conversely, a monostable front can never be pinned, it gives rise to a secondary pulse past the defect and we calculate the time this pulse takes to appear. These radically different behaviors of bistable and monostable fronts raise issues for modelers in particular areas of biology, as for example, the study of tumor growth in the presence of different tissues.

nlin.PS↗

arXiv2020

Berry phase in superconducting multiterminal quantum dots

We report on the study of the non-trivial Berry phase in superconducting multiterminal quantum dots biased at commensurate voltages. Starting with the time-periodic Bogoliubov-de Gennes equations, we obtain a tight binding model in the Floquet space, and we solve these equations in the semiclassical limit. We observe that the parameter space defined by the contact transparencies and quartet phase splits into two components with a non-trivial Berry phase. We use the Bohr-Sommerfeld quantization to calculate the Berry phase. We find that if the quantum dot level sits at zero energy, then the Berry phase takes the values $φ_B=0$ or $φ_B=π$. We demonstrate that this non-trivial Berry phase can be observed by tunneling spectroscopy in the Floquet spectra. Consequently, the Floquet-Wannier-Stark ladder spectra of superconducting multiterminal quantum dots are shifted by half-a-period if $φ_B=π$. Our numerical calculations based on Keldysh Green's functions show that this Berry phase spectral shift can be observed from the quantum dot tunneling density of states.

cond-mat.mes-hall↗

arXiv2020

Coupling Conditions for Water Waves at Forks

We considered the propagation of nonlinear shallow water waves in a narrow channel presenting a fork. We aimed at computing the coupling conditions for a 1D effective model, using 2D simulations and an analysis based on the conservation laws. For small amplitudes, this analysis justifies the well-known Stoker interface conditions, so that the coupling does not depend on the angle of the fork. We also find this in the numerical solution. Large amplitude solutions in a symmetric fork also tend to follow Stoker's relations, due to the symmetry constraint. For non symmetric forks, 2D effects dominate so that it is necessary to understand the flow inside the fork. However, even then, conservation laws give some insight in the dynamics.

physics.class-ph↗

arXiv2020

Nonlinear waves in networks: a simple approach using the sine-Gordon equation

To study the propagation of nonlinear waves across Y- and T-type junctions, we consider the 2D sine--Gordon equation as a model and study the dynamics of kinks and breathers in such geometries. The comparison of the energies reveals that the angle of the fork plays no role. Motivated by this, we introduce a 1D effective equation whose solutions agree well with the 2D simulations for kink and breather solutions. For branches of equal width, breather crossing occurs approximately when $v > 1 - ω$, where $v$ is the breather celerity and $ω$ is its frequency. We then characterize the breathers in the two upper branches by estimating their velocity and frequency. These new breathers are slower than the initial breather and up-shifted in frequency. In perspective, this study could be generalized to more complex nonlinear waves.

nlin.PS↗

arXiv2019

Engineering the Floquet spectrum of superconducting multiterminal quantum dots

Here we present a theoretical investigation of the Floquet spectrum in multiterminal quantum dot Josephson junctions biased with commensurate voltages. We first draw an analogy between the electronic band theory and superconductivity which enlightens the time-periodic dynamics of the Andreev bound states. We then show that the equivalent of the Wannier-Stark ladders observed in semiconducting superlattices via photocurrent measurements, appears as specific peaks in the finite frequency current fluctuations of superconducting multiterminal quantum dots. In order to probe the Floquet-Wannier-Stark ladder spectra, we have developed an analytical model relying on the sharpness of the resonances. The charge-charge correlation function is obtained as a factorized form of the Floquet wave-function on the dot and the superconducting reservoir populations. We confirm these findings by Keldysh Green's function calculations, in particular regarding the voltage and frequency dependence of the resonance peaks in the current-current correlations. Our results open up a road-map to quantum correlations and coherence in the Floquet dynamics of superconducting devices.

cond-mat.mes-hall↗

arXiv2018

Scattering of a short electromagnetic pulse from a Lorentz-Duffing film: theoretical and numerical analysis

We combine scattering theory, Fourier, traveling wave and asymptotic analyses together with numerical simulations to present interesting and practically useful properties of femtosecond pulse interaction with thin films. The dispersive material is described by a single resonance Lorentz model and its nonlinear extension with a cubic Duffing-type nonlinearity. A key feature of the Lorentz dielectric function is that its real part becomes negative between its zero and its pole, generating a forbidden region. We illustrate numerically the linear interaction of the pulse with the film using both scattering theory and Fourier analysis. Outside this region we show the generation of a sequence of pulses separated by round trips in the Fabry-Perot cavity due to multiple reflections. When the pulse spectrum is inside the forbidden region, we observe total reflection. Near the pole of the dielectric function, we demonstrate the slowing down of the pulse (group velocity tending to zero) in the medium that behaves as a high-Q cavity. We use the combination of analysis and simulations in the linear regime to validate the delta function approximation of the thin layer; this collapses the forbidden region to a single resonant point of the spectrum. We also study the single cycle pulse interaction with a thin film and show three distinct types of reflection: half-pulse, sinusoidal wave train and cosine wavelet. Finally we analyze the influence of a strong nonlinearity and observe that the film switches from reflecting to trasparent.

physics.optics↗

arXiv2018

Wave dynamics on networks: method and application to the sine-Gordon equation

We consider a scalar Hamiltonian nonlinear wave equation formulated on networks; this is a non standard problem because these domains are not locally homeomorphic to any subset of the Euclidean space. More precisely, we assume each edge to be a 1D uniform line with end points identified with graph vertices. The interface conditions at these vertices are introduced and justified using conservation laws and an homothetic argument. We present a detailed methodology based on a symplectic finite difference scheme together with a special treatment at the junctions to solve the problem and apply it to the sine-Gordon equation. Numerical results on a simple graph containing four loops show the performance of the scheme for kinks and breathers initial conditions.

math-ph↗

arXiv2017

Periodic orbits in nonlinear wave equations on networks

We consider a cubic nonlinear wave equation on a network and show that inspecting the normal modes of the graph, we can immediately identify which ones extend into nonlinear periodic orbits. Two main classes of nonlinear periodic orbits exist: modes without soft nodes and others. For the former which are the Goldstone and the bivalent modes, the linearized equations decouple. A Floquet analysis was conducted systematically for chains; it indicates that the Goldstone mode is usually stable and the bivalent mode is always unstable. The linearized equations for the second type of modes are coupled, they indicate which modes will be excited when the orbit destabilizes. Numerical results for the second class show that modes with a single eigenvalue are unstable below a treshold amplitude. Conversely, modes with multiple eigenvalues seem always unstable. This study could be applied to coupled mechanical systems.

nlin.PS↗