English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The tunneling effect predicted by B.Josephson (Nobel Prize, 1973) concernsthe 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 differentialequations on 2-torus depending on 3 parameters: $B$ (abscissa), $A$ (ordinate),$\omega$ (frequency). We study its rotation number $\rho(B,A;\omega)$ as afunction of $(B,A)$ with fixed $\omega$. The phase-lock areas are the levelsets $L_r:=\{\rho=r\}$ with non-empty interiors; they exist for $r\in\mathbb Z$(Buchstaber, Karpov, Tertychnyi). Each $L_r$ is an infinite chain of domainsgoing 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 itspunctured neighborhood in the vertical line lies in $\operatorname{Int}(L_r)$(confirming another conjecture). We first prove deformability of eachconstriction to another one, with arbitrarily small $\omega$, of the same$\rho$, $\ell:=\frac B\omega$ and type (positive or not), using equivalentdescription of model by linear systems of differential equations on$\bar{\mathbb C}$ (Buchstaber, Karpov, Tertychnyi) and studying theirisomonodromic deformations described by Painlev\'e 3 equations. Thennon-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 resultsand elaborated methods and open problems.

On families of constrictions in model of overdamped Josephson junction and Painlev\'e 3 equation