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.Comment: 11 page

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.Comment: Meccanica (2018). arXiv admin note: text overlap with arXiv:1602.0907

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 {\epsilon}, is investigated. Results obtained prove that for slow time {\epsilon}t < 1 waves are propagated almost undisturbed, while for fast time t > 1 {\epsilon} diffusion effects prevail.Comment: Ricerche di Matematica (2018

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

Towards soliton solutions of a perturbed sine-Gordon equation

We give arguments for the existence of {\it exact} travelling-wave (in particular solitonic) solutions of a perturbed sine-Gordon equation on the real line or on the circle, and classify them. The perturbation of the equation consists of a constant forcing term and a linear dissipative term. Such solutions are allowed exactly by the energy balance of these terms, and can be observed experimentally e.g. in the Josephson effect in the theory of superconductors, which is one of the physical phenomena described by the equation.Comment: 16 pages, 4 figures include

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