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BIMSA Integrable Systems Seminar
Model of Josephson junction, dynamical systems on $T^2$, isomonodromic deformations and Painleve 3 equations
Model of Josephson junction, dynamical systems on $T^2$, isomonodromic deformations and Painleve 3 equations
Organizers
Speaker
Glutsyuk Alexey
Time
Tuesday, April 16, 2024 4:00 PM - 5:00 PM
Venue
Online
Online
Zoom 873 9209 0711
(BIMSA)
Abstract
The tunneling effect predicted by B.Josephson (Nobel Prize, 1973) concerns the Josephson junction: two superconductors separated by a arrow 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 $\bar{\mathbb{C}}$ (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.
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 $\bar{\mathbb{C}}$ (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.