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 TMF, 2015, Volume 182, Number 3, Pages 373–404 (Mi tmf8770)

Holomorphic solutions of the double confluent Heun equation associated with the RSJ model of the Josephson junction

V. M. Buchstabera, S. I. Tertychnyib

a Steklov Mathematical Institute, RAS, Moscow, Russia
b All-Russian Scientific Research Institute for Physical and Radio-Technical Measurements (VNIIFTRI), Mendeleevo, Moscow Oblast, Russia

Abstract: This work is a continuation of research on a first-order nonlinear differential equation applied in the overshunted model of the Josephson junction. The approach is based on the relation between this equation and the double confluent Heun equation, which is a second-order linear homogeneous equation with two irregular singular points. We describe the conditions on the equation parameters under which its general solution is an analytic function on the Riemann sphere except at $0$ and $\infty$. We construct an explicit basis of the solution space. One of the functions in this basis is regular everywhere except $0$, and the other is regular everywhere except $\infty$. We show that in the framework of the RSJ model of Josephson junction dynamics, the described situation corresponds to the condition that the Shapiro step vanishes if all the solutions of the double confluent Heun equation are single-valued on the Riemann sphere without $0$ and $\infty$.

Keywords: double confluent Heun equation, holomorphic solution, dynamical system on a torus with the identical Poincaré map

 Funding Agency Grant Number Russian Foundation for Basic Research 14-01-00506

DOI: https://doi.org/10.4213/tmf8770

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English version:
Theoretical and Mathematical Physics, 2015, 182:3, 329–355

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Revised: 06.10.2014

Citation: V. M. Buchstaber, S. I. Tertychnyi, “Holomorphic solutions of the double confluent Heun equation associated with the RSJ model of the Josephson junction”, TMF, 182:3 (2015), 373–404; Theoret. and Math. Phys., 182:3 (2015), 329–355

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tmf8770
• https://doi.org/10.4213/tmf8770
• http://mi.mathnet.ru/eng/tmf/v182/i3/p373

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. M. Buchstaber, S. I. Tertychnyi, “On a Remarkable Sequence of Bessel Matrices”, Math. Notes, 98:5 (2015), 714–724
2. V. M. Buchstaber, S. I. Tertychnyi, “Automorphisms of the solution spaces of special double-confluent Heun equations”, Funct. Anal. Appl., 50:3 (2016), 176–192
3. Buchstaber V.M., Glutsyuk A.A., “On determinants of modified Bessel functions and entire solutions of double confluent Heun equations”, Nonlinearity, 29:12 (2016), 3857–3870
4. V. M. Buchstaber, A. A. Glutsyuk, “On monodromy eigenfunctions of Heun equations and boundaries of phase-lock areas in a model of overdamped Josephson effect”, Proc. Steklov Inst. Math., 297 (2017), 50–89
5. V. M. Buchstaber, S. I. Tertychnyi, “Representations of the Klein Group Determined by Quadruples of Polynomials Associated with the Double Confluent Heun Equation”, Math. Notes, 103:3 (2018), 357–371
6. Glutsyuk A.A., “On Constrictions of Phase-Lock Areas in Model of Overdamped Josephson Effect and Transition Matrix of the Double-Confluent Heun Equation”, J. Dyn. Control Syst., 25:3 (2019), 323–349
7. A. V. Malyutin, “The Rotation Number Integer Quantization Effect in Braid Groups”, Proc. Steklov Inst. Math., 305 (2019), 182–194
8. S. I. Tertychnyi, “Solution space monodromy of a special double confluent Heun equation and its applications”, Theoret. and Math. Phys., 201:1 (2019), 1426–1441
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