Solution Space Monodromy of a Special Double Confluent Heun Equation and Its Applications
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.
Keywordsdouble confluent Heun equation solution space automorphism monodromy composition law matrix representation solution continuation RSJ model of Josephson junction
Unable to display preview. Download preview PDF.
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.
- 4.G. Bor, M. Levi, R. Perline, and S. Tabachnikov, “Tire tracks and integrable curve evolution,” arXiv: 1705.06314v3 [math.DG] (2017).Google 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
- 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
- 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
- 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
- 27.Y. Bibilo, “Josephson effect and isomonodromic deformations,” arXiv:1805.11759v2 [math.CA] (2018).Google 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