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

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.

Keywords: изомонодромные деформациибыстро-медленные системычисло вращенияrotation numberisomonodromic deformationsslow-fast systemsдинамическая система на тореphase-lock areasзоны фазового захватаdynamical system on torusmodel of Josephson junctionlinear systems of differential equations on the Riemann spherePainleve 3 equationмодель Джозефсоновского переходалинейные системы дифференциальных уравнений на сфере Риманауравнение Пенлеве 3

Bibilo Y., Glutsyuk A., Nonlinearity 2022 Vol. 35 No. 10 P. 5427-5480

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 ...

Added: December 20, 2022

Glutsyuk A., Journal of Dynamical and Control Systems 2019 Vol. 25 No. 3 P. 323-349

In 1973, B. Josephson received a Nobel Prize for discovering a new fundamentaleffect 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 nonlinear ...

Added: August 20, 2018

Schurov I., Труды Московского математического общества 2010 № 71 С. 200-234

В типичных быстро-медленных системах на двумерном торе с единственным параметром при сколь угодно малом значении этого параметра существуют притягивающие уточные циклы. Это существенно отличает динамику на торе от динамики аналогичных систем на плоскости. Ранее это было показано для систем с выпуклой медленной кривой. В настоящей работе рассматриваются системы с невыпуклой медленной кривой. Получены верхняя и ...

Added: May 14, 2014

Ilya Schurov, Nikita Solodovnikov, / Cornell University. Series math "arxiv.org". 2014. No. 1405.3251.

Slow-fast systems on the two-torus are studied. As it was shown before, canard cycles are generic in such systems, which is in drastic contrast with the planar case. It is known that if the rotation number of the Poincare map is integer and the slow curve is connected, the number of canard limit cycles is ...

Added: May 14, 2014

Бухштабер В. М., Glutsyuk A., Труды Математического института им. В.А. Стеклова РАН 2017 Т. 297 С. 62-104

Abstract—We study a family of double confluent Heun equations of the form LE = 0, where
L = L(λ,μ,n) is a family of second-order differential operators acting on germs of holomorphic
functions of one complex variable. They depend on complex parameters λ, μ, and n. The
restriction of the family to real parameters satisfying the inequality λ + μ^2>0 ...

Added: June 29, 2018

A. Klimenko, O. Romaskevich, Moscow Mathematical Journal 2014 Vol. 14 No. 2 P. 367-384

A three-parametrical family of ODEs on a torus arises from a model of Josephson effect in a resistive case when a Josephson junction is biased by a sinusoidal microwave current. We study asymptotics of Arnold tongues of this family on the parametric plane (the third parameter is fixed) and prove that the boundaries of the ...

Added: September 5, 2014

Romaskevich O. L., Клепцын В. А., Schurov I., Наноструктуры. Математическая физика и моделирование 2013 Т. 8 № 1 С. 31-46

In order to model the processes taking place in systems with Josephson contacts, a differential equation on a torus with three parameters is used. One of the parameters of the system can be considered small and the methods of the fast-slow systems theory can be applied. The properties of the phase-lock areas - the subsets ...

Added: December 25, 2012

On determinants of modified Bessel functions and entire solutions of double confluent Heun equations

Buchstaber V.M., Glutsyuk A. A., Nonlinearity 2016 Vol. 29 No. 12 P. 3857-3870

We investigate the question on existence of entire solutions of well-known linear differential equations that are linearizations of nonlinear equations modeling the Josephson effect in superconductivity. We consider the modified Bessel functions Ij(x) of the first kind, which are Laurent series coefficients of the analytic function family . For every we study the family parametrized by , , of -matrix functions formed ...

Added: June 17, 2021

Schurov I., Клепцын В. А., Romaskevich O. L., Наноструктуры. Математическая физика и моделирование 2014 Т. 8 № 1 С. 31-46

In order to model the processes taking place in systems with Josephson contacts, a differential equation on a torus with three parameters is used. One of the parameters of the system can be considered small and the methods of the fast-slow systems theory can be applied. The properties of the phase-lock areas – the subsets ...

Added: December 25, 2014

Gavrylenko P., Marshakov A., / Cornell University. Series "Working papers by Cornell University". 2015. No. 1507.08794.

We consider the conformal blocks in the theories with extended conformal W-symmetry for the integer Virasoro central charges. We show that these blocks for the generalized twist fields on sphere can be computed exactly in terms of the free field theory on the covering Riemann surface, even for a non-abelian monodromy group. The generalized twist ...

Added: October 14, 2015

Schurov I., Solodovnikov N., Journal of Dynamical and Control Systems 2017 Vol. 23 No. 3 P. 481-498

Slow-fast systems on the two-torus are studied. As it was shown before, canard cycles are generic in such systems, which is in drastic contrast with the planar case. It is known that if the rotation number of the Poincaré map is an integer and the slow curve is connected, the number of canard limit cycles ...

Added: July 17, 2016

Gavrylenko P., Lisovyy O., / Cornell University. Series math-ph "arXiv". 2016. No. 1608.00958.

We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with n regular singular points on the Riemann sphere and generic monodromy in GL(N,ℂ). The corresponding operator acts in the direct sum of N(n−3) copies of L2(S1). Its kernel has a block integrable form and is expressed in terms of fundamental solutions of ...

Added: September 20, 2016

Buff X., Goncharuk Nataliya, Journal of Modern Dynamics 2015 Vol. 9 P. 169-190

We investigate the notion of complex rotation number which was introduced by V.I.Arnold in 1978. Let f: R/Z -> R/Z be a (real) analytic orientation preserving circle diffeomorphism and let omega in C/Z be a parameter with positive imaginary part. Construct a complex torus by glueing the two boundary components of the annulus { z ...

Added: October 10, 2013

Schurov I., Клепцын В. А., Romaskevich O. L., Наноструктуры. Математическая физика и моделирование 2013 Т. 8 № 1 С. 31-46

In order to model the processes taking place in systems with Josephson contacts, a differential equation on a torus with three parameters is used. One of the parameters of the system can be considered small and the methods of the fast-slow systems theory can be applied. The properties of the phase-lock areas – the subsets ...

Added: December 17, 2014

Gavrylenko P., Iorgov N., Lisovyy O., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2018 Vol. 14 P. 1-27

We derive Fredholm determinant and series representation of the tau function of the Fuji-Suzuki-Tsuda system and its multivariate extension, thereby generalizing to higher rank the results obtained for Painlevé VI and the Garnier system. A special case of our construction gives a higher rank analog of the continuous hypergeometric kernel of Borodin and Olshanski. We also ...

Added: November 22, 2018

Poberezhny V. A., / ИТЭФ. Series "Препринты ИТЭФ". 2014. No. 50.14.

We prove that any non-resonant Fuchsian system with commutative monodromy is in fact a commutative system, that is a system with commuting residues. For logarithmic connection that Fuchsian system presents that implies the triviality of its isomonodromic deformations. ...

Added: March 26, 2015

Poberezhny V. A., / ИТЭФ. Series "Препринты ИТЭФ". 2012. No. 57/12.

In this work we investigate the action of generalized Schlesinger transformation on the isomonodromic families of meromorphic connections on the linear bundles of rank two and degree zero over an elliptic curve. The main interest is the action of the gauge transformation on the moduli space of vector bundles. the central result is the explicit ...

Added: March 31, 2014

Glutsyuk A., Netay I. V., Journal of Dynamical and Control Systems 2020 Vol. 26 P. 785-820

The paper deals with a three-parameter family of special dou- ble confluent Heun equations that was introduced and studied by V. M. Buchstaber and S. I. Tertychnyi as an equivalent presentation of a model of overdamped Josephson junction in superconductivity. The parameters are l, λ, μ ∈ R. Buchstaber and Tertychnyi described those parameter values, for which the ...

Added: October 19, 2020

Kazakov A., Lerman L. M., Kulagin N., Mathematical Modeling of Natural Phenomena 2013 Vol. 8 No. 5 P. 155-172

We demonstrate that a piecewise linear slow-fast Hamiltonian system with an equilibrium of the saddle-center type can have a sequence of small parameter values for which a one-round homoclinic orbit to this equilibrium exists. This contrasts with the well-known findings by Amick and McLeod and others that solutions of such type do not exist in ...

Added: March 29, 2015

Gurevich E., Сяинова Д. Т., Журнал Средневолжского математического общества 2014 Т. 16 № 2 С. 46-56

We specify S. Batterson's results of [7] where classes of isotopic maps on torus contained Morse-Smale diffeomorphisms are described. Following to ideas of paper [7], we describe isotopic classes, contained gradient-like diffeomorphisms, present all admitted types of periodic data of such diffeomorphisms and provide an algorithm of realization of each type of periodic data. ...

Added: October 14, 2014

V. A. Poberezhny, Journal of Mathematical Sciences 2013 Vol. 195 No. 4 P. 533-540

We consider systems of linear differential equations discussing some classical and modern results in the Riemann problem, isomonodromic deformations, and other related topics. Against this background, we illustrate the relations between such phenomena as the integrability, the isomonodromy, and the Painlevé property. The recent advances in the theory of isomonodromic deformations presented show perfect agreement ...

Added: February 14, 2014

Gavrylenko P., Lisovyy O., Communications in Mathematical Physics 2018 Vol. 363 No. 1 P. 1-58

We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with n regular singular points on the Riemann sphere and generic monodromy in GL (N,ℂ). The corresponding operator acts in the direct sum of N (n − 3) copies of L2 (S1). Its kernel has a block integrable form and is expressed in ...

Added: September 12, 2018

Zabrodin A., Zotov A., Journal of Mathematical Physics 2012 Vol. 53 No. 7 P. 073507-1-073507-19

The Painlevé-Calogero correspondence is extended to auxiliary linear problems associated with Painlevé equations. The linear problems are represented in a new form which has a suggestive interpretation as a "quantized" version of the Painlevé-Calogero correspondence. Namely, the linear problem responsible for the time evolution is brought into the form of non-stationary Schrödinger equation in imaginary ...

Added: September 19, 2012

Levin A., Ольшанецкий М. А., Зотов А. В., Успехи математических наук 2014 Т. 69 № 1(415) С. 39-124

В данной работе изомонодромные задачи описываются в терминах плоских G-расслоений на проколотых эллиптических кривых Σ_τ и связностей с регулярными особенностями в отмеченных точках. Расслоения классифицируются по их характеристическим классам, которые являются элементами группы вторых когомологий H^2(Σ_τ,Z(G)), где Z(G) – центр G. По каждой простой комплексной группе Ли G и произвольному характеристическому классу определяется пространство модулей ...

Added: January 21, 2015