Geodesic
On the Monodromy-Preserving Deformation of a Double Confluent Heun Equation
Informatics and Automation, Topology, Geometry, Combinatorics, and Mathematical Physics, Tome 326 (2024), pp. 330-367 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the second-order linear differential equation presented in Subsection 4.8 of the paper by J. Dereziński, A. Ishkhanyan, and A. Latosiński [SIGMA 17, 056 (2021)] and called there the deformed double confluent Heun equation. We prove that the additional singular point arising during the construction of the equation does not affect the analytic structure of its solution space. Moreover, we prove that under certain conditions a one-parameter transformation of the equation, called a deformation, with coefficients expressed in terms of the third Painlevé transcendent, leaves its monodromy unchanged. The proofs are self-contained.
Mots-clés : double confluent Heun equation, nondestructive singular point, Frobenius norm, third Painlevé transcendent, transport equation
Keywords: deformed double confluent Heun equation, isomonodromicity. 
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     author = {S. I. Tertichniy},
     title = {On the {Monodromy-Preserving} {Deformation} of a {Double} {Confluent} {Heun} {Equation}},
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S. I. Tertichniy. On the Monodromy-Preserving Deformation of a Double Confluent Heun Equation. Informatics and Automation, Topology, Geometry, Combinatorics, and Mathematical Physics, Tome 326 (2024), pp. 330-367. http://geodesic.mathdoc.fr/item/TRSPY_2024_326_a14/

[1] Bibilo Y., Glutsyuk A.A., “On families of constrictions in model of overdamped Josephson junction and Painlevé 3 equation”, Nonlinearity, 35:10 (2022), 5427–5480 | DOI | MR | Zbl

[2] Clarkson P.A., “Painlevè equations—nonlinear special functions”, J. Comput. Appl. Math., 153:1–2 (2003), 127–140 | DOI | MR | Zbl

[3] Dereziński J., Ishkhanyan A., Latosiński A., “From Heun class equations to Painlevé equations”, SIGMA. Symmetry, Integrability Geom. Methods Appl., 17 (2021), 056 | DOI | MR | Zbl

[4] Gromak V.I., Laine I., Shimomura S., Painlevé differential equations in the complex plane, W. de Gruyter, Berlin, 2002 | DOI | MR

[5] Handbook of mathematical functions with formulas, graphs, and mathematical tables, NBS Appl. Math. Ser., 55, ed. by M. Abramowitz, I.A. Stegun, Natl. Bureau Stand., Washington, 1964 | MR

[6] Jimbo M., Miwa T., Ueno K., “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I: General theory and $\tau $-function”, Physica D, 2:2 (1981), 306–352 | DOI | MR | Zbl

[7] Jimbo M., Miwa T., “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II”, Physica D, 2:3 (1981), 407–448 | DOI | MR | Zbl

[8] Jimbo M., Miwa T., “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III”, Physica D, 4:1 (1981), 26–46 | DOI | MR | Zbl

[9] McCumber D.E., “Effect of ac impedance on dc voltage–current characteristics of superconductor weak-link junctions”, J. Appl. Phys., 39:7 (1968), 3113–3118 | DOI

[10] Renne M.J., Polder D., “Some analytical results for the resistively shunted Josephson junction”, Rev. phys. appl., 9:1 (1974), 25–28 | DOI

[11] Schmidt D., Wolf G., “Double confluent Heun equation”, Heun's differential equations, Part C, ed. by A. Ronveaux, Oxford Univ. Press, Oxford, 1995, 129–188

[12] V. V. Schmidt, Introduction to the Physics of Superconductors, 2nd ed., MTsNMO, Moscow, 2000 | DOI | Zbl

[13] The Physics of Superconductors: Introduction to Fundamentals and Applications, Springer, Berlin, 1997 | DOI | Zbl

[14] Shapiro S., Janus A.R., Holly S., “Effect of microwaves on Josephson currents in superconducting tunneling”, Rev. Mod. Phys., 36:1 (1964), 223–225 | DOI

[15] Slavyanov S.Yu., “Painlevé equations as classical analogues of Heun equations”, J. Phys. A: Math. Gen., 29:22 (1996), 7329–7335 | DOI | MR | Zbl

[16] S. Yu. Slavyanov, “Structural theory of special functions”, Theor. Math. Phys., 119:1 (1999), 393–406 | DOI | DOI | MR | Zbl

[17] S. Yu. Slavyanov, “Isomonodromic deformations of Heun and Painlevé equations”, Theor. Math. Phys., 123:3 (2000), 744–753 | DOI | DOI | MR | Zbl

[18] Slavyanov S., Stesik O., “Antiquantization as a specific way from the Statistical physics to the Regular physics”, Physica A, 521 (2019), 512–518 | DOI | MR | Zbl

[19] Stewart W.C., “Current–voltage characteristic of Josephson junction”, Appl. Phis. Lett., 12:8 (1968), 277–280 | DOI

[20] Tertychniy S.I., Long-term behavior of solutions of the equation $\dot \phi +\sin \phi =f$ with periodic $f$ and the modeling of dynamics of overdamped Josephson junctions, E-print, 2005, arXiv: math-ph/0512058 | DOI