Dynamics of point Josephson junctions in a microstrip line

TL;DR

This paper models the electrodynamics of point Josephson junctions in a superconducting cavity using a wave equation with nonlinearities, revealing how resonances influence the system's behavior through spectral analysis.

Contribution

It introduces a novel long wave model with Dirac delta nonlinearities and a spectral problem to analyze resonances in Josephson junction arrays.

Findings

Resonances are characterized by a specific spectral problem.

At resonances, the system simplifies to two ordinary differential equations.

The model provides insights into current-voltage characteristics of Josephson junction arrays.

Abstract

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.