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

Abstract

Abstract The tunnelling effect predicted by 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 modelled by a family of differential equations on two-torus depending on three parameters: B (abscissa), A (ordinate), ω (frequency). We study its rotation number ρ ( B , A ; ω ) as a function of ( B , A ) with fixed ω . The phase-lock areas are the level sets L r ≔ { ρ = r } with non-empty interiors; they exist for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>r</mml:mi> <mml:mo>∈</mml:mo> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:math> (Buchstaber, Karpov, Tertychnyi). Each L r is an infinite chain of domains going vertically to infinity and separated by points. Those separating points for which A ≠ 0 are called constrictions . We show that: (1) all the constrictions in L r lie on the axis { B = ωr }; (2) each constriction is positive : this means that some its punctured neighbourhood on the axis { B = ωr } lies in Int( L r ). These results confirm experiments by physicists (1970ths) and two mathematical conjectures. We first prove deformability of each constriction to another one, with arbitrarily small ω and the same <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>ℓ</mml:mi> <mml:mo>≔</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ω</mml:mi> </mml:mrow> </mml:mfrac> </mml:math> , using equivalent description of model by linear systems of differential equations on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> </mml:math> (Buchstaber, Karpov, Tertychnyi) and studying their isomonodromic deformations described by Painlevé 3 equations. Then non-existence of ghost constrictions (i.e., constrictions either with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>ρ</mml:mi> <mml:mo>≠</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ω</mml:mi> </mml:mrow> </mml:mfrac> </mml:math> , or of non-positive type) with a given ℓ for small ω is proved by slow-fast methods.

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