Model of Josephson junction, dynamical systems on $\mathbb T^2$, isomonodromic deformations and Painleve 3 equations

Abstract: The tunneling effect predicted by B.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 modeled by a family of differential equations on 2-torus depending on 3 parameters: $B$ (abscissa), $A$ (ordinate), $\omega$ (frequency). We study its rotation number $\rho(B,A;\omega)$ as a function of $(B,A)$ with fixed $\omega$. The phase-lock areas are those level sets $L_r:=\{\rho=r\}$ that have non-empty interiors. They exist only for integer rotation number values $r$: this is the rotation number quantization effect discovered by Buchstaber, Karpov and Tertychnyi. They are analogues of the famous Arnold tongues. Each $L_r$ 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 $\omega=2$, 1, 0.3 at the presentation.

We show that: 1) all constrictions in $L_r$ lie in the vertical line $\{ B=\omega r\}$; 2) each constriction is positive, that is, some its punctured neighborhood in the vertical line lies in $\operatorname{Int}(L_r)$. These results, obtained in collaboration with Yulia Bibilo, confirm experiences of physicists (pictures from physical books of 1970-th) and two mathematical conjectures.

The proof uses an equivalent description of model by linear systems of differential equations on $\oc$ (found by Buchstaber, Karpov and Tertychnyi), their isomonodromic deformations described by Painleve 3 equations and methods of the theory of slow-fast systems.

If the time allows we will discuss new results and open questions.

mathematical physicsdynamical systemsquantum algebrarepresentation theorysymplectic geometry

Audience: general audience

( slides | video )

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BIMSA Integrable Systems Seminar

Series comments: The aim is to bring together experts in integrable systems and related areas of theoretical and mathematical physics and mathematics. There will be research presentations and overview talks.

Audience: Graduate students and researchers interested in integrable systems and related mathematical structures, such as symplectic and Poisson geometry and representation theory.

The zoom link will be distributed by mail, so please join the mailing list if you are interested in attending the seminar.

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