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

This article is cited in 11 scientific papers (total in 11 papers)

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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Received: 15.07.2014
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
\by V.~M.~Buchstaber, S.~I.~Tertychnyi
\paper Holomorphic solutions of the~double confluent Heun equation associated with the~RSJ model of the~Josephson junction
\jour TMF
\yr 2015
\vol 182
\issue 3
\pages 373--404
\jour Theoret. and Math. Phys.
\yr 2015
\vol 182
\issue 3
\pages 329--355

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    This publication is cited in the following articles:
    1. Glutsyuk A.A., Netay I.V., “On Spectral Curves and Complexified Boundaries of the Phase-Lock Areas in a Model of Josephson Junction”, J. Dyn. Control Syst.  crossref  mathscinet  isi
    2. V. M. Buchstaber, S. I. Tertychnyi, “On a Remarkable Sequence of Bessel Matrices”, Math. Notes, 98:5 (2015), 714–724  mathnet  crossref  crossref  mathscinet  isi  elib
    3. 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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    4. 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  crossref  mathscinet  zmath  isi  elib  scopus
    5. 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  mathnet  crossref  crossref  mathscinet  isi  elib
    6. 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  mathnet  crossref  crossref  mathscinet  isi  elib
    7. A. A. Glutsyuk, “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  crossref  isi
    8. A. V. Malyutin, “The Rotation Number Integer Quantization Effect in Braid Groups”, Proc. Steklov Inst. Math., 305 (2019), 182–194  mathnet  crossref  crossref  mathscinet  isi  elib
    9. 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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    10. S. I. Tertychniy, “Symmetries of the space of solutions to special double confluent Heun equations of integer order”, J. Math. Phys., 60:10 (2019), 103501  crossref  mathscinet  isi
    11. V. M. Buchstaber, S. I. Tertychnyi, “Categories of Symmetry Groups of the Space of Solutions of the Special Doubly Confluent Heun Equation”, Math. Notes, 110:5 (2021), 643–654  mathnet  crossref  crossref
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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