Seminar on Analysis. Speaker: Alexey Glutsyuk

Abstract.
In 1962 B. Josephson predicted a tunnelling effect arising in a system of two superconductors separated by a thin dielectric layer, now known as the Josephson junction. It involves the existence of a supercurrent crossing the junction and governed by equations discovered by Josephson.

The overdamped Josephson junction is modeled by a family of differential equations on the two-dimensional torus, known as the RSJ model, depending on three parameters: the abscissa B, the ordinate A, and a fixed frequency of external forcing. The corresponding rotation number is a function of (B,A). The phase-lock areas are those level sets of the rotation number that have non-empty interior; they exist only for integer rotation numbers. This quantization phenomenon was discovered by V. M. Buchstaber, O. V. Karpov, and S. I. Tertychnyi, who showed that the system is equivalent to a family of second-order linear differential equations on the Riemann sphere: special double confluent Heun equations.

The phase-lock areas form garlands of infinitely many domains separated by points. Those separation points that do not lie on the abscissa axis are called constrictions. In joint work with Yu. P. Bibilo, the speaker showed that in each phase-lock area all constrictions lie on a single vertical line whose abscissa equals the product of the forcing frequency and the rotation number. The proof uses the above equivalence with linear equations, Stokes phenomena, isomonodromic deformations governed by Painlevé III equations, holomorphic vector bundles in the spirit of A. A. Bolibruch’s work, and methods from slow–fast system theory. Similar methods were recently used by the speaker to compute the genus of spectral curves associated with special double confluent Heun equations admitting polynomial solutions.

A. S. Gorsky asked whether there exist dynamical systems on the torus equivalent to general Heun equations with four singularities. A family of such systems, obtained as a deformation of the RSJ model, was recently constructed by the speaker. In this deformed model all constrictions disappear. This is joint work with A. A. Alexandrov.

In the talk, we sketch proofs of the above results and give a survey of related developments and open questions.

We warmly invite you to attend the meeting of the seminar. Please feel free to share this announcement with colleagues and students who might be interested.

We look forward to your participation!

Best regards,
Alexander Menovschikov
on behalf of the organizing committee

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