# Solution Space Monodromy of a Special Double Confluent Heun Equation and Its Applications

## Abstract

We consider three linear operators determining automorphisms of the solution space of a special double confluent Heun equation of positive integer order (L-operators). We propose a new method for describing properties of the solution space of this equation based on using eigenfunctions of one of the L-operators, called the universal L-operator. We construct composition laws for L-operators and establish their relation to the monodromy transformation of the solution space of the special double confluent Heun equation. We find four functionals quadratic in eigenfunctions of the universal automorphism; they have a property with respect to the considered equation analogous to the property of the first integral. Based on them, we construct matrix representations of the L-operators and also the monodromy operator. We give a method for extending solutions of the special double confluent Heun equation from the subset *Re* z > *0* of a complex plane to a maximum domain on which the solution exists. As an example of its application to the RSJ model theory of overdamped Josephson junctions, we give the explicit form of the transformation of the phase difference function induced by the monodromy of the solution space of the special double confluent Heun equation and propose a way to continue this function from a half-period interval to any given interval in the domain of the function using only algebraic transformations.

## Keywords

double confluent Heun equation solution space automorphism monodromy composition law matrix representation solution continuation RSJ model of Josephson junction## Preview

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## Notes

### Acknowledgments

The author is grateful to V. M. Buchstaber for the useful discussions of the paper.

**Conflicts of interest.** The author declares no conflicts of interest.

## References

- 1.R. Foote, “Geometry of the Prytz planimeter,”
*Rep. Math. Phys.*,**42**, 249–271 (1998); arXiv:math/9808070v1 (1998).ADSMathSciNetCrossRefGoogle Scholar - 2.M. Levi and S. Tabachnikov, “On bicycle tire tracks geometry, hatchet planimeter, Menzin’s conjecture, and oscillation of unicycle tracks,”
*Experiment. Math.*,**18**, 173–186 (2009).MathSciNetCrossRefGoogle Scholar - 3.R. L. Foote, M. Levi, and S. Tabachnikov, “Tractrices, bicycle tire tracks, hatchet planimeters, and a 100-year-old conjecture,”
*Amer. Math. Monthly*,**120**, 199–216 (2013).MathSciNetCrossRefGoogle Scholar - 4.G. Bor, M. Levi, R. Perline, and S. Tabachnikov, “Tire tracks and integrable curve evolution,” arXiv: 1705.06314v3 [math.DG] (2017).Google Scholar
- 5.J. Guckenheimer and Yu. S. Ilyashenko, “The duck and the devil: Canards on the staircase,”
*Moscow Math. J.*,**1**, 27–47 (2001).MathSciNetCrossRefGoogle Scholar - 6.V. A. Kleptsyn, O. L. Romaskevich, and I. V. Shchurov, “Josephson effect and fast–slow systems [in Russian],”
*Nanostrukt. Matem. Fiz. i Modelir.*,**8**, 31–46 (2013).Google Scholar - 7.W. C. Stewart, “Current–voltage characteristics of Josephson junctions,”
*Appl. Phys. Lett.*,**12**, 277–280 (1968).ADSCrossRefGoogle Scholar - 8.D. E. McCumber, “Effect of ac impedance on dc voltage–current characteristics of superconductor weak-link junctions,”
*J. Appl. Phys.*,**39**, 3113–3118 (1968).ADSCrossRefGoogle Scholar - 9.P. Mangin and R. Kahn,
*Superconductivity: An Introduction*, Springer, New York (2017).CrossRefGoogle Scholar - 10.V. M. Buchstaber, O. V. Karpov, and S. I. Tertychnyi, “Rotation number quantization effect,”
*Theor. Math. Phys.*,**162**, 211–221 (2010).MathSciNetCrossRefGoogle Scholar - 11.Yu. S. Ilyashenko, D. A. Ryzhov, and D. A. Filimonov, “Phase-lock effect for equations modeling resistively shunted Josephson junctions and for their perturbations,”
*Funct. Anal. Appl.*,**45**, 192–203 (2011).MathSciNetCrossRefGoogle Scholar - 12.A. Glutsyuk and L. Rybnikov, “On families of differential equations on two-torus with all phase-lock areas,”
*Nonlinearity*,**30**, 61–72 (2017).ADSMathSciNetCrossRefGoogle Scholar - 13.G. V. Osipov and A. V. Polovinkin,
*Synchronization by an External Periodic Action*[in Russian], Nizhny Novgorod State Univ. Press, Nizhny Novgorod (2005).Google Scholar - 14.V. M. Buchstaber and A. A. Glutsyuk, “Josephson effect, Arnold tongues, and double confluent Heun equations,” Talk at Intl. Conf. “Contemporary Mathematics,” dedicated to the 80th birthday of V. I. Arnold, Higher School of Economics, Skolkovo Inst. of Science and Technology, Steklov Math. Inst., Moscow, 18–23 December 2017 (2017).Google Scholar
- 15.S. I. Tertychniy, “Long-term behavior of solutions to the equation
*ø*+sin*ø*=*f*with periodic*f*and the modeling of dynamics of overdamped Josephson junctions: Unlectured notes,” arXiv:math-ph/0512058v1 (2005).Google Scholar - 16.V. M. Buchstaber and S. I. Tertychnyi, “Dynamical systems on a torus with identity Poincaré map which are associated with the Josephson effect,”
*Russian Math. Surveys*,**69**, 383–385 (2014).ADSMathSciNetCrossRefGoogle Scholar - 17.S. I. Tertychniy, “The interrelation of the special double confluent Heun equation and the equation of RSJ model of Josephson junction revisited,” arXiv:1811.03971v1 [math-ph] (2018).Google Scholar
- 18.D. Schmidt and G. Wolf, “Double confluent Heun equation,” in:
*Heun’s Diffrential Equations*(A. Ronveaux, ed.), Oxford Univ. Press, Oxford (1995), pp. 129–188.Google Scholar - 19.S. Yu. Slavyanov and W. Lay,
*Special Functions: A Unified Theory Based on Singularities*, Oxford Univ. Press, Oxford (2000).zbMATHGoogle Scholar - 20.The Heun Project, “Heun functions, their generalizations and applications: Bibliography,” http://theheunproject.org/bibliography.html (2017).Google Scholar
- 21.M. Horta¸csu, “Heun functions and some of their applications in physics,”
*Adv. High Energy Phys.*,**2018**, 8621573 (2018); arXiv:1101.0471v11 [math-ph] (2011).Google Scholar - 22.V. M. Buchstaber and A. A. Glutsyuk, “On phase-lock areas in a model of Josephson effect and double confluent Heun equations,” Talk at Intl. Conf. “Real and Complex Dynamical Systems,” dedicated to the 75th anniversary of Yu. S. Il’yashenko, Steklov Math. Inst., Moscow, 30 November 2018 (2018).zbMATHGoogle Scholar
- 23.V. M. Buchstaber and S. I. Tertychnyi, “Explicit solution family for the equation of the resistively shunted Josephson junction model,”
*Theor. Math. Phys.*,**176**, 965–986 (2013).MathSciNetCrossRefGoogle Scholar - 24.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**, 323–349 (2019).MathSciNetCrossRefGoogle Scholar - 25.V. M. Buchstaber and S. I. Tertychnyi, “Holomorphic solutions of the double confluent Heun equation associated with the RSJ model of the Josephson junction,”
*Theor. Math. Phys.*,**182**, 329–355 (2015).MathSciNetCrossRefGoogle Scholar - 26.V. M. Buchstaber and A. A. Glutsyuk, “On determinants of modified Bessel functions and entire solutions of double confluent Heun equations,”
*Nonlinearity*,**29**, 3857–3870 (2016); arXiv:1509.01725v4 [math.DS] (2015).ADSMathSciNetCrossRefGoogle Scholar - 27.Y. Bibilo, “Josephson effect and isomonodromic deformations,” arXiv:1805.11759v2 [math.CA] (2018).Google Scholar
- 28.A. A. Glutsyuk, V. A. Kleptsyn, D. A. Filimonov, and I. V. Shchurov, “On the adjacency quantization in an equation modeling the Josephson effect,”
*Funct. Anal. Appl.*,**48**, 272–285 (2014).MathSciNetCrossRefGoogle Scholar - 29.V. M. Buchstaber and 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**, 50–89 (2017).MathSciNetCrossRefGoogle Scholar - 30.A. A. Salatic and S. Yu. Slavyanov, “Antiquantization of the double confluent Heun equation: The Teukolsky equation,”
*Russ. J. Nonlinear Dyn.*,**15**, 79–85 (2019).MathSciNetzbMATHGoogle Scholar - 31.S. Yu. Slavyanov, “Isomonodromic deformations of Heun and Painlevé equations,”
*Theor. Math. Phys.*,**123**, 744–753 (2000).MathSciNetCrossRefGoogle Scholar - 32.S. Yu. Slavyanov and O. L. Stesik, “Antiquantization of deformed Heun-class equations,”
*Theor. Math. Phys.*,**186**, 118–125 (2016).MathSciNetCrossRefGoogle Scholar - 33.S. I. Tertychniy, “Square root of the monodromy map for the equation of RSJ model of Josephson junction,” arXiv:1901.01103v3 [math.CA] (2019).Google Scholar
- 34.V. M. Buchstaber and S. I. Tertychnyi, “Automorphisms of the solution spaces of special double-confluent Heun equations,”
*Funct. Anal. Appl.*,**50**, 176–192 (2016).MathSciNetCrossRefGoogle Scholar - 35.V. M. Buchstaber and S. I. Tertychnyi, “Representations of the Klein group determined by quadruples of polynomials associated with the double confluent Heun equation,”
*Math. Notes*,**103**, 357–371 (2018).MathSciNetCrossRefGoogle Scholar