On Constrictions of Phase-Lock Areas in Model of Overdamped Josephson Effect and Transition Matrix of the Double-Confluent Heun Equation

Abstract

In 1973 B.Josephson received Nobel Prize for discovering a new fundamental effect concerning a Josephson junction, -a system of two superconductors separated by a very narrow dielectric: there could exist a supercurrent tunneling through this junction. We will discuss the model of the overdamped Josephson junction, which is given by a family of first order non-linear ordinary differential equations on two-torus depending on three parameters: a fixed parameter ω (the frequency); a pair of variable parameters (B, A) that are called respectively the abscissa, and the ordinate. It is important to study the rotation number of the system as a function ρ = ρ(B, A) and to describe the phase-lock areas: its level sets L r = {ρ = r} with non-empty interiors. They were studied by V.M.Buchstaber, O.V.Karpov, S.I.Tertychnyi, who observed in 2010 that the phase-lock areas exist only for integer values of the rotation number. It is known that each phase-lock area is a garland of infinitely many bounded domains going to infinity in the vertical direction; each two subsequent domains are separated by one point, which is called constriction (provided that it does not lie in the abscissa axis). Those points of intersection of the boundary ∂L r of the phaselock area L r with the line Λ r = {B = rω} (which is called its axis) that are not constrictions are called simple intersections. It is known that our family of dynamical systems is related to appropriate family of double confluent Heun equations with the same parameters via Buchtaber-Tertychnyi construction. Simple intersections correspond

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

“…Positivity of constrictions was stated there as [29, conjecture 1.13]. It was also shown in [29] that Theorem 1.7 would imply Theorem 1.4.…”

“…It was proved in [28, theorem 1.2] that for every r ∈ Z the constrictions in L r have abscissas B = ω, ∈ Z, ≡ r(mod 2), ∈ [0, r]. For further results and discussion of this conjecture see [13, section 5] and [28,29,30]. 1c)], see also [20, p. 331].…”

“…Transversal regularity of the fibration by level sets ρ(B, A) = const / ∈ Z with fixed ω on the complement to the union of the phase-lock areas was proved in [13, proposition 5.3]. Conjectures on alignment and positivity of constrictions (now Theorems 1.4 and 1.7 respectively) were stated respectively in [28] and [29] and studied respectively in [28,29] and [29], where some partial results were obtained. Theorem 1.4 for ω ≥ 1 was proved in [28].…”

On families of constrictions in the model of an overdamped Josephson junction

“…Для любой частоты ω все перемычки зон захвата положительны. Теорема 2 подтверждает гипотезу из [5], где доказано, что нейтральных перемычек нет. Перемычки, либо имеющие абсциссу B = ρω, либо неположительные, будем называть призрачными.…”

“…типа Джозефсона на сфере Римана (см. [3] и библ., а также [4,5]). Исследуются их изомонодромные деформации в 4-мерном семействе т.н.…”