Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Monica De Angelis

TL;DR

The paper surveys mathematical contributions to Josephson junction dynamics, focusing on the perturbed sine-Gordon framework and the role of dissipative and memory effects. It develops a unifying third-order parabolic operator ${\cal L}=\varepsilon \partial_{xxt}-\partial_{tt}+\partial_{xx}-\alpha \partial_t$ and its integro-differential variant ${\cal L}_R$ to connect diverse physical models, including ESJJ and WJJ geometries. Through explicit fundamental solutions, Green-function representations, Fourier-series analyses, and integral-equation formulations, it establishes existence, uniqueness, a priori estimates, and long-time asymptotics, showing that initial disturbances decay and boundary effects remain bounded. The work highlights open problems such as interface conditions for window junctions, boundary condition generalizations, and extensions to nonconstant coefficients, pointing to rich theory with practical implications for superconducting devices.

Abstract

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.

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

TL;DR

The paper surveys mathematical contributions to Josephson junction dynamics, focusing on the perturbed sine-Gordon framework and the role of dissipative and memory effects. It develops a unifying third-order parabolic operator and its integro-differential variant to connect diverse physical models, including ESJJ and WJJ geometries. Through explicit fundamental solutions, Green-function representations, Fourier-series analyses, and integral-equation formulations, it establishes existence, uniqueness, a priori estimates, and long-time asymptotics, showing that initial disturbances decay and boundary effects remain bounded. The work highlights open problems such as interface conditions for window junctions, boundary condition generalizations, and extensions to nonconstant coefficients, pointing to rich theory with practical implications for superconducting devices.

Abstract

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.

Paper Structure

This paper contains 5 sections, 2 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Mathematical models and equivalences
  3. Mathematical results
  4. Open problems
  5. Conclusion

Key Result

Theorem 3.1

The function $\ K \,$ has the same basic properties of the fundamental solution of the heat equation, that is: $\,\,K(x,t) \, \, \in C ^ {\infty} \,\,\,\,$ for $\,\,\, t>0, \,\,\,\, x \,\,\, \in \Re.$ For fixed $\, t\,>\,0,\,\,\, K \,$ and its derivatives are exponentially vanishing as fast as $\ For $\,t\,>\,0,\,$ it is $\,\,\, {\cal L}_R\,K =\, 0. \,\,$ Moreover, it results

Theorems & Definitions (2)

  • Theorem 3.1
  • Theorem 3.2