On Germs of Constriction Curves in Model of Overdamped Josephson Junction, Dynamical Isomonodromic Foliation and Painlevé 3 Equation
Résumé
B.Josephson (Nobel Prize, 1973) predicted a tunnelling effect for a system of two superconductors separated by a narrow dielectric (such a system is called Josephson junction): 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), ω (frequency). We study its rotation number ρ(B, A; ω) as a function of parameters. The three-dimensional phase-lock areas are the level sets L r := {ρ = r} ⊂ R 3 with non-empty interiors; they exist for r ∈ Z (Buchstaber, Karpov, Tertychnyi). For every fixed ω > 0 and r ∈ Z the planar slice L r ∩(R 2 B,A ×{ω}) is a garland of domains going vertically to infinity and separated by points; those separating points for which A = 0 are called constrictions. In a joint paper by Yu.Bibilo and the author, it was shown that 1) at each constriction the rescaled abscissa := B ω is integer and = ρ; 2) the family Constr of constrictions with given ∈ Z is an analytic submanifold in (R 2 +) a,s , a = ω-1 , s = A ω. In the present paper we show that 1) the limit points of Constr are β ,k = (0, s ,k), where s ,k are the positive zeros of the-th Bessel function J (s); 2) to each β ,k accumulates exactly one its component C ,k (constriction curve), and it lands at β ,k regularly. Known numerical phase-lock area pictures show that high components of interior of each phase-lock area L r look similar. In his paper with Bibilo, the author introduced a candidate to the self-similarity map between neighbor components: the Poincaré map of the dynamical isomonodromic foliation governed by Painlevé 3 equation. Whenever well-defined, it preserves the rotation number function. We show that the Poincaré map is well-defined on a neighborhood of the plane {a = 0} ⊂ R 2 ,a × (R +) s , and it sends each constriction curve germ (C ,k , β ,k) to (C ,k+1 , β ,k+1).
Domaines
Mathématiques [math]
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