arXiv subjects

Monica De Angelis

Publications and source records attributed to Monica De Angelis.

At least 19 recordsLinked to original sources

arXiv2025

Mathematical Contributions to the Dynamics of the Josephson Junctions: State of the Art and Open Problems

Mathematical models related to some Josephson junctions are pointed out and attention is drawn to the solutions of certain initial boundary problems and to some of their estimates. In addition, results of rigorous analysis of the behaviour of these solutions when the time tends to infinity and when the small parameter tends to zero are cited. These analyses lead us to mention some of the open problems.

math-ph↗

arXiv2023

Hydrodynamic regime and cold plasmas hit by short laser pulses

We briefly report and elaborate on some conditions allowing a hydrodynamic description of the impact of a very short and arbitrarily intense laser pulse onto a cold plasma, as well as the localization of the first wave-breaking due to the plasma inhomogeneity. We use a recently developed fully relativistic plane model whereby we reduce the system of the Lorentz-Maxwell and continuity PDEs into a 1-parameter family of decoupled systems of non-autonomous Hamilton equations in dimension 1, with the light-like coordinate $ξ=ct\!-\!z$ replacing time $t$ as an independent variable. Apriori estimates on the Jacobian $\hat J$ of the change from Lagrangian to Eulerian coordinates in terms of the input data (initial density and pulse profile) are obtained applying Liapunov direct method to an associated family of pairs of ODEs; wave-breaking is pinpointed by the inequality $\hat J\le 0$. These results may help in drastically simplifying the study of extreme acceleration mechanisms of electrons, which have very important applications.

physics.plasm-ph↗

arXiv2023

Hopf bifurcations in dynamics of excitable systems

A general FitzHugh-Rinzel model, able to describe several neuronal phenomena, is considered. Linear stability and Hopf bifurcations are investigated by means of the spectral equation for the ternary autonomous dynamical system and the analysis is driven by both an admissible critical point and a parameter which characterizes the system.

nlin.CD↗

arXiv2023

Transport Phenomena in Excitable Systems: Existence of Bounded Solutions and Absorbing Sets

In this paper, the transport phenomena of synaptic electric impulses are considered. The FitzHugh--Nagumo and FitzHugh--Rinzel models appear mathematically appropriate for evaluating these scientific issues. Moreover, applications of such models arise in several biophysical phenomena in different fields such as, for instance, biology, medicine and electronics, where, by means of nanoscale memristor networks, scientists seek to reproduce the behavior of biological synapses. The present article deals with the properties of the solutions of the FitzHugh--Rinzel system in an attempt to achieve, by means of a suitable ``energy function'', conditions ensuring the boundedness and existence of absorbing sets in the phase space. The results obtained depend on several parameters characterizing the system, and, as an example, a concrete case is considered.

math-ph↗

arXiv2022

Hydrodynamic impacts of short laser pulses on plasmas

We determine conditions allowing to simplify the description of the impact of a short and arbitrarily intense laser pulse onto a cold plasma at rest. If both the initial plasma density and pulse profile have plane simmetry, then suitable matched upper bounds on the maximum and the relative variations of the initial density, as well as the intensity and duration of the pulse, ensure a strictly hydrodynamic evolution of the electron fluid (without wave-breaking or vacuum-heating) during its whole interaction with the pulse, while ions can be regarded as immobile. We use a recently developed fully relativistic plane model whereby the system of the (Lorentz-Maxwell and continuity) PDEs is reduced into a family of highly nonlinear but decoupled systems of non-autonomous Hamilton equations with one degree of freedom, with the light-like coordinate $ξ=ct\!-\!z$ instead of time $t$ as an independent variable, and new apriori estimates (eased by use of a Liapunov function) of the solutions in terms of the input data (initial density and pulse profile). If the laser spot radius $R$ is finite but not too small the same conclusions hold for the part of the plasma close to the axis $\vec{z}$ of cylindrical symmetry. These results may help in drastically simplifying the study of extreme acceleration mechanisms of electrons.

physics.plasm-ph↗

arXiv2022

A priori estimates for solutions of FitzHugh-Rinzel system

The FitzHugh-Rinzel system is able to describe some biophysical phenomena, such as bursting oscillations, and the study of its solutions can help to better understand several behaviours of the complex dynamics of biological systems. We express the solutions by means of an integral equation involving the fundamental solution $ H(x,t) $ related to a non linear integro-differential equation. Properties of $ H(x,t) $ allow us to obtain a priori estimates for solutions determined in the whole space, showing both the influence of the initial data and the source term.

math.AP↗

arXiv2022

On solutions related to FitzHugh-Rinzel type model

A ternary autonomous dynamical system of FitzHugh-Rinzel type is analyzed. The system, at start, is reduced to a nonlinear integro differential equation. The fundamental solution $ H(x,t) $ is explicitly determined and the initial value problem is analyzed in the whole space. The solution is expressed by means of an integral equation involving $ H(x,t) $. Moreover, adding an extra control term, explicit solutions are achieved.

math.AP↗

arXiv2022

A note on explicit solutions of FitzHugh-Rinzel system

The numerous scientific feedbacks that the FitzHugh-Rinzel system (FHR) is having in various scientific fields, lead to further studies on the determination of its explicit solutions. Indeed, such a study can help to get a better understanding of several behaviors in the complex dynamics of biological systems. In this note, a class of traveling wave solutions is determined and specific solutions are achieved to explicitly show the contribution due to a diffusion term considered in the FHR model.

nlin.AO↗

arXiv2018

A wave equation perturbed by viscous terms: fast and slow times diffusion effects in a Neumann problem

A Neumann problem for a wave equation perturbed by viscous terms with small parameters is considered. The interaction of waves with the diffusion effects caused by a higher-order derivative with small coefficient ε, is investigated. Results obtained prove that for slow time εt < 1 waves are propagated almost undisturbed, while for fast time t > 1 ε diffusion effects prevail.

math-ph↗

arXiv2018

On the transition from parabolicity to hyperbolicity for a nonlinear equation under Neumann boundary conditions

An integro differential equation which is able to describe the evolution of a large class of dissipative models, is considered. By means of an equivalence, the focus shifts to the perturbed sine- Gordon equation that in superconductivity finds interesting applications in multiple engineering areas. The Neumann boundary problem is considered, and the behaviour of a viscous term, defined by a high order derivative with small diffusion coefficient , is investigated. The Green function, expressed by means of Fourier series, is considered, and an estimate is achieved. Furthermore, some classes of solutions of the hyperbolic equation are determined, proving that there exists at least one solution with bounded derivatives. Results obtained prove that diffusion effects are bounded and tend to zero when e tends to zero.

math-ph↗

arXiv2016

On diffusion effects of the perturbed sine-Gordon equation with Neumann boundary conditions

The Neumann boundary problem for the perturbed sine-Gordon equation describing the electrodynamics of Josephson junctions has been considered. The behavior of a viscous term, described by a higher-order derivative with small diffusion coefficient, is investigated. The Green function related to the linear third order operator is determined by means of Fourier series, and properties of rapid convergence are established. Furthermore, some classes of solutions of the hyperbolic equation have been determined, proving that there exists at least one solution whose derivatives are bounded. Results prove that diffusion effects are bounded and tend to zero when the small diffusion coefficient tends to zero.

math-ph↗

arXiv2014

On asymptotic effects of boundary perturbations in exponentially shaped Josephson junctions

A parabolic integro differential operator L, suitable to describe many phenomena in various physical fields, is considered. By means of equivalence between L and the third order equation describing the evolution inside an exponentially shaped Josephson junction (ESJJ), an asymptotic analysis for (ESJJ) is achieved, explicitly evaluating, boundary contributions related to the Dirichlet problem.

math-ph↗

arXiv2013

On a parabolic operator of dissipative systems

A parabolic integro differential operator operator L suitable to describe many phenomena in various physical fields,is considered. By means of equivalence between L and the third order equation which describe the evolution inside an exponentially shaped Josephson junction (ESJJ), an asymptotic analysis for (ESJJ) is achieved, evaluating explicitly boundary contributions related to the Dirichlet problem.

math-ph↗

arXiv2012

On a non linear third - order parabolic equation

Aim of this paper is the qualitative analysis of the solution of a boundary value problem for a third-order non linear parabolic equation which describes several dissipative models. When the source term is linear, the problem is explictly solved by means of a Fourier series with properties of rapid convergence. In the non linear case,appropriate estimates of this series allow to deduce the asymptotic behaviour of the solution.

math-ph↗

arXiv2012

Parabolic - hyperbolic boundary layer

A boundary value problem related to a parabolic higher order operator with a small parameter is analized. When the small parameter tends to zero, the reduced operator is hyperbolic. When t tends to infinity a parabolic hyperbolic boundary layer appears. In this paper a rigorous asymptotic approximation uniformly valid for all t is established.

math-ph↗

arXiv2012

Existence and uniqueness of solutions of a class of 3rd order dissipative problems with various boundary conditions describing the Josephson effect

We prove existence and uniqueness of solutions of a large class of initial-boundary-value problems characterized by a quasi-linear third order equation (the third order term being dissipative) on a finite space interval with Dirichlet, Neumann or pseudoperiodic boundary conditions. The class includes equations arising in superconductor theory, such as a well-known modified sine-Gordon equation describing the Josephson effect, and in the theory of viscoelastic materials.

math-ph↗

arXiv2012

Diffusion and wave behaviour in linear Voigt model

A boundary value problem related to a third- order parabolic equation with a small parameter is analized. This equation models the one-dimensional evolution of many dissipative media as viscoelastic fluids or solids, viscous gases, superconducting materials, incompressible and electrically conducting fluids. Moreover, the third-order parabolic operator regularizes various non linear second order wave equations. In this paper, the hyperbolic and parabolic behaviour of the solution is estimated by means of slow time and fast time. As consequence, a rigorous asymptotic approximation for the solution is established.

math-ph↗

arXiv2012

On a Model of Superconductivity and Biology

The paper deals with a semilinear integrodifferential equation that characterizes several dissipative models of Viscoelasticity, Biology and Superconductivity. The initial - boundary problem with Neumann conditions is analyzed. When the source term F is a linear function, then the explicit solution is obtained. When F is non linear, some results on existence, uniqueness and a priori estimates are deduced. As example of physical model the reaction - diffusion system of Fitzhugh Nagumo is considered.

q-bio.NC↗