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Working paper

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

math. arxive. Cornell University, 2021. No. 2011.07839.
Glutsyuk A., Bibilo Y.
We study family of dynamical systems on 2-torus modeling over-damped Josephson junction in superconductivity. It depends on three parameters (B,A;ω): B (abscissa), A(ordinate), ω (a fixed frequency).We study the rotation numberρ(B,A;ω) as a function of (B,A) withfixedω. Aphase-lock areais the level set Lr:={ρ=r}, if it has an on-empty interior. This holds for r∈Z (a result by V.M.Buchstaber, O.V.Karpov and S.I.Tertychnyi). It is known that each phase-lock area is an infinite garland of domains going to infinity in the vertical direction and separated by points called constrictions (expect for the separation points with A= 0). We show that all the constrictions in Lr lie in its axis {B=ωr}, confirming an experimental fact (conjecture) observed numerically by S.I.Tertychnyi, V.A.Kleptsyn, D.A.Filimonov, I.V.Schurov. We prove that each constriction is positive: the phase-lock area germ contains the vertical line germ (confirming another conjecture). To do this, we study family of linear systems on the Riemann sphere equivalently describing the model: the Josephson type systems.We study their Jimbo isomonodromic deformations described by solutions of Painleve 3 equations. Using results of this study and a Riemann–Hilbert approach, we show that each constriction can be analytically deformed to constrictions with the same l:=Bω and arbitrarily small ω. Then non-existence of ”ghost” constrictions (nonpositive or with ρ not equal to l) with a given l for small ω is proved by slow-fast methods.