On families of constrictions in model of overdamped Josephson junction and Painlevé 3 equation

Abstract

The tunneling effect predicted by B.Josephson (Nobel Prize, 1973) concerns the Josephson junction: two superconductors separated by a narrow dielectric. It states existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modeled by a family of differential equations on 2-torus depending on 3 parameters: $B$ (abscissa), $A$ (ordinate), $\omega$ (frequency). We study its rotation number $\rho (B,A;\omega )$ as a function of $(B,A)$ with fixed $\omega$. The phase-lock areas are the level sets $L_r:=\{\rho =r\}$ with non-empty interiors; they exist for $r\in \mathbb{Z}$ (Buchstaber, Karpov, Tertychnyi). Each $L_r$ is an infinite chain of domains going vertically to infinity and separated by points called constrictions (expect for those with $A=0$). We show that: 1) all the constrictions in $L_r$ lie in its axis $\{B=\omega r\}$ (confirming a conjecture of Tertychnyi, Kleptsyn, Filimonov, Schurov); 2) each constriction is positive: some its punctured neighborhood in the vertical line lies in $\mathrm{Int}(L_r)$ (confirming another conjecture). We first prove deformability of each constriction to another one, with arbitrarily small $\omega$, of the same $\rho$, $\ell :=\frac{B}{\omega }$ and type (positive or not), using equivalent description of model by linear systems of differential equations on $\bar{\mathbb{C}}$ (Buchstaber, Karpov, Tertychnyi) and studying their isomonodromic deformations described by Painlevé 3 equations. Then non-existence of ghost constrictions (i.e., constrictions either with $\rho \ne \ell$, or of non-positive type) with a given $\ell$ for small $\omega$ is proved by slow-fast methods. In Section 6 we present applications of results and elaborated methods and open problems.

73 pages, 10 figures