# Moscow Mathematical Journal

Volume 23, Issue 4, October–December 2023 pp. 479–513.

On Germs of Constriction Curves in Model of Overdamped Josephson Junction, Dynamical Isomonodromic Foliation and Painlevé 3 Equation

**Authors**:
Alexey Glutsyuk (1)

**Author institution:**(1) CNRS, UMR 5669 (UMPA, ENS de Lyon), France;

HSE University, Moscow, Russia;

Kharkevich Institute for Information Transmission Problems (IITP, RAS), Moscow

**Summary: **

B. Josephson (Nobel
Prize, 1973) predicted a *tunnelling effect* for a system of two superconductors
separated by a narrow dielectric (such a system is called *Josephson junction*): 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 parameters.
The *three-dimensional phase-lock areas* are the level sets $L_r:=\{\rho=r\}\subset\mathbb{R}^3$ 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 integer and $\ell=\rho$; 2) the family $\mathrm{Constr}_\ell$ of constrictions with given $\ell\in\mathbb{Z}$ is an analytic submanifold in
$(\mathbb{R}^2_+)_{a,s}$, $a=\omega^{-1}$, $s=\frac A\omega$.
In the present paper we show that 1) the limit points of $\mathrm{Constr}_\ell$ are $\beta_{\ell,k}=(0,s_{\ell,k})$, where
$s_{\ell,k}$ are the positive zeros of the $\ell$-th Bessel function $J_\ell(s)$; 2) to each
$\beta_{\ell,k}$ accumulates exactly one its component $\mathcal{C}_{\ell,k}$ (constriction curve), and it lands at $\beta_{\ell,k}$ regularly.
Known numerical phase-lock area pictures show that high components of interior of each phase-lock area $L_r$ look similar. In his paper with Bibilo,
the author introduced a candidate to the self-similarity map between neighbor components: the Poincaré map of the dynamical isomonodromic foliation
governed by Painlevé 3 equation. Whenever well defined, it preserves the
rotation number function. We show that the Poincaré 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 each constriction curve
germ $(\mathcal{C}_{\ell,k},\beta_{\ell,k})$ to $(\mathcal{C}_{\ell,k+1},\beta_{\ell,k+1})$.

2020 Math. Subj. Class. 34M03, 34A26, 34E15

**Keywords:**Josephson junction, differential equations on torus, rotation number, phase-lock areas, linear systems of complex differential equations, monodromy operator, Stokes matrices, isomonodromic deformations, Painlevé 3 equation.

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