Model of Josephson junction, dynamical systems on T2, isomonodromic deformations and Painleve 3 equations | BlMSA Integrable Systems Seminar

Abstract

The tunneling effect predicted by B.Josephson (Nobel Prize, 1973) concerns the Josephsoniunction: two superconductors separated by a arrow dielectric. lt states existence of a supercurrentthrough it and equations governing it. The overdamped Josephson junction is modeled by a familyof diferential equations on 2-torus depending on 3 parameters: B (abscissa),A (ordinate), w(frequency). We study its rotation number p(B, A; w) as a function of (B, A) with fixed w. Thephase-lock areas are those level sets Lr :=fp=r, that have non-empty interiors. They existonly for integer rotation number values r: this is the rotation number quantization effect discoveredby Buchstaber, Karpov and Tertychnyi. They are analogues of the famous Arnold tongues. Each L,is an infinite chain of domains going vertically to infinity and separated by points called constrictions(expect for those with A=0). See the phase-lock area portraits for w=2, 1, 0.3 at thepresentation. We show that: 1) all constrictions in L. lie in the vertical line fB=wrl; 2) eachconstriction is positive, that is, some its punctured neighborhood in the vertical line lies in Int(L.)These results, obtained in collaboration with Yulia Bibilo, confirm experiences of physicists (picturesfrom physical books of 1970-th) and two mathematical conjectures. The proof uses an equivalentdescription of model by linear systems of diferential equations on C (found by Buchstaber, Karpovand Tertychnyi), their isomonodromic deformations described by Painleve 3 equations and methodsof the theory of slow-fast systems. lf the time allows we wil discuss new results and openquestions.