On germs of constriction curves in model of overdamped Josephson junction, dynamical isomonodromic foliation and Painlev\'e 3 equation

TL;DR

This paper investigates the structure of constriction curves in the parameter space of overdamped Josephson junctions, revealing their relation to Bessel function zeros and the isomonodromic foliation governed by Painlevé 3, with implications for phase-lock areas.

Contribution

It establishes the limiting behavior of constriction points in the parameter space and connects the Poincaré map of the isomonodromic foliation to these constrictions, advancing understanding of the junction's dynamics.

Findings

Limit points of constriction curves are zeros of Bessel functions.

The Poincaré map is well-defined near a specific plane and shifts constriction points systematically.

High components of phase-lock areas exhibit self-similar structures.

Abstract

B.Josephson (Nobel Prize, 1973) predicted tunnelling effect for a system (called Josephson junction) of two superconductors separated by a narrow dielectric: 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: $B$, $A$, $\omega$. We study its rotation number $\rho(B,A;\omega)$ as a function of parameters. The three-dimensional 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). For every fixed $\omega>0$ and $r\in\mathbb Z$ the planar slice $L_r\cap(\mathbb R^2_{B,A}\times\{\omega\})$ is a garland of domains going vertically to infinity and separated by points; those separating points for which $A\neq0$ are called constrictions. In a joint paper by Yu.Bibilo and the author, it was shown that 1) at each constriction the rescaled abscissa $\ell:=\frac B\omega$ is equal to $\rho$; 2) the family of constrictions with given $\ell\in\mathbb Z$ is an analytic submanifold $Constr_\ell$ in $(\mathbb R^2_+)_{a,s}$, $a=\omega^{-1}$, $s=\frac A\omega$. Here we show that the limit points of $Constr_\ell$ are $\beta_{\ell,k}=(0,s_{\ell,k})$, where $s_{\ell,k}>0$ are zeros of the Bessel function $J_\ell(s)$, and it lands at them regularly. Known numerical pictures show that high components of $Int(L_r)$ look similar. In his paper with Bibilo, the author introduced a candidate to the self-similarity map between neighbor components: the Poincar\'e map of the dynamical isomonodromic foliation governed by Painlev\'e 3 equation. Whenever well-defined, it preserves $\rho$. We show that the Poincar\'e map is well-defined on a neighborhood of the plane $\{ a=0\}\subset\mathbb R^2_{\ell,a}\times(\mathbb R_+)_s$, and it sends $\beta_{\ell,k}$ to $\beta_{\ell,k+1}$ for integer $\ell$.