Abstract:
We consider a family of differential equations on a torus modeling Josefson's effect from the superconductivity theory. We investigate level sets of the rotation number that have a non-empty interior.
In contrast to the classical situation, such zones occur only for integer values of the rotation number (proved by V. M. Buchstaber, O. V. Karpov, S. I. Tertychnyi and later by Yu. S. Ilyashenko). Every such zone is a sequence of plane domains separeted by jumpers.
The considered family is equivalent to a family of linear differential equations of the second order on the Riemann sphere: biconfluent Heun equations.
In the talk I will give a review of results and open questions on the geometry of the domains defined above. In particular, I will describe coordinates of jumpers (following Buchstaber, Tertychnyi, Kleptsyn, Filimonov, Shchurov and Glutsyuk). A special attention will be payed to a recent paper by Glutsyuk and Buchstaber, where we obtain a new result on determinants formed by modified Bessel functions and use this to prove a hypothesis of Buchstaber and Tertychnyi concerning a partial description of ordinates of jumpers.
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