On families of constrictions in model of overdamped Josephson junction and Painlev\'e 3 equation
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 $\operatorname{Int}(L_r)$ (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 Painlev\'e 3 equations. Then non-existence of ghost constrictions (i.e., constrictions either with $\rho\neq\ell$, 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.