Abstract:
We construct a solution of an analogue of the Schrödinger equation for the Hamiltonian H1(z,t,q1,q2,p1,p2) corresponding to the second equation P21 in the Painlevé I hierarchy. This solution is obtained by an explicit change of variables from a solution of systems of linear equations whose compatibility condition is the ordinary differential equation P21 with respect to z. This solution also satisfies an analogue of the Schrödinger
equation corresponding to the Hamiltonian H2(z,t,q1,q2,p1,p2) of a Hamiltonian system with respect to t compatible with P21. A similar situation occurs for the P22 equation in the Painlevé II hierarchy.
Citation:
B. I. Suleimanov, ““Quantizations” of Higher Hamiltonian Analogues of the Painlevé I and Painlevé II Equations with Two Degrees of Freedom”, Funktsional. Anal. i Prilozhen., 48:3 (2014), 52–62; Funct. Anal. Appl., 48:3 (2014), 198–207
\Bibitem{Sul14}
\by B.~I.~Suleimanov
\paper ``Quantizations'' of Higher Hamiltonian Analogues of the Painlev\'e I and Painlev\'e II Equations with Two Degrees of Freedom
\jour Funktsional. Anal. i Prilozhen.
\yr 2014
\vol 48
\issue 3
\pages 52--62
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\jour Funct. Anal. Appl.
\yr 2014
\vol 48
\issue 3
\pages 198--207
\crossref{https://doi.org/10.1007/s10688-014-0061-0}
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https://doi.org/10.4213/faa3150
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This publication is cited in the following 14 articles:
V. A. Pavlenko, “Solutions of Analogs of Time-Dependent Schrödinger
Equations Corresponding to a Pair of H2+2+1
Hamiltonian Systems in the Hierarchy of Degenerations
of an Isomonodromic Garnier System”, Diff Equat, 60:1 (2024), 77
V. A Pavlenko, “REShENIYa ANALOGOV VREMENNYKh URAVNENIY ShR¨EDINGERA, SOOTVETSTVUYuShchIKh PARE GAMIL'TONOVYKh SISTEM ????2+2+1 IERARKhII VYROZhDENIY IZOMONODROMNOY SISTEMY GARN'E”, Differencialʹnye uravneniâ, 60:1 (2024), 76
Dan Dai, Wen-Gao Long, “Asymptotics and Total Integrals of the \(\textrm{P}_{\textrm I}^2\) Tritronquée Solution and Its Hamiltonian”, SIAM J. Math. Anal., 56:4 (2024), 5350
V. A. Pavlenko, “Solutions of the analogues of time-dependent Schrödinger equations corresponding to a pair of H3+2 Hamiltonian systems”, Theoret. and Math. Phys., 212:3 (2022), 1181–1192
A. V. Domrin, M. A. Shumkin, B. I. Suleimanov, “Meromorphy of solutions for a wide class of ordinary differential equations of Painlevé type”, Journal of Mathematical Physics, 63:2 (2022)
V. V. Tsegel'nik, “Properties of solutions of two second-order differential equations with the Painlevé property”, Theoret. and Math. Phys., 206:3 (2021), 315–320
B. I. Suleimanov, A. M. Shavlukov, “Integrable Abel equation and asymptotics
of symmetry solutions of Korteweg-de Vries equation”, Ufa Math. J., 13:2 (2021), 99–106
B. I. Suleimanov, “Isomonodromic quantization of the second Painlevé equation by means of conservative Hamiltonian systems with two degrees of freedom”, St. Petersburg Math. J., 33:6 (2022), 995–1009
Adler V.E., “Nonautonomous Symmetries of the Kdv Equation and Step-Like Solutions”, J. Nonlinear Math. Phys., 27:3 (2020), 478–493
V. I. Kachalov, Yu. S. Fedorov, “O metode malogo parametra v nelineinoi matematicheskoi fizike”, Sib. elektron. matem. izv., 15 (2018), 1680–1686
V. A. Pavlenko, B. I. Suleimanov, “Solutions to analogues of non-stationary Schrödinger equations defined by isomonodromic Hamilton system H2+1+1+1”, Ufa Math. J., 10:4 (2018), 92–102
V. A. Pavlenko, B. I. Suleimanov, ““Quantizations” of isomonodromic Hamilton system H72+1”, Ufa Math. J., 9:4 (2017), 97–107
D. P. Novikov, B. I. Suleimanov, ““Quantization” of an isomonodromic Hamiltonian Garnier system with two degrees of freedom”, Theoret. and Math. Phys., 187:1 (2016), 479–496
B. I. Suleimanov, “Quantum aspects of the integrability of the third Painlevé equation and a non-stationary time Schrödinger equation with the Morse potential”, Ufa Math. J., 8:3 (2016), 136–154
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