Abstract:
We construct a solution of an analogue of the Schrödinger equation for the Hamiltonian $ H_1 (z, t, q_1, q_2, p_1, p_2) $ corresponding to the second equation $P_1^2$ 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 $P_1^2$ with respect to $z$. This solution also satisfies an analogue of the Schrödinger
equation corresponding to the Hamiltonian $ H_2 (z, t, q_1, q_2, p_1, p_2) $ of a Hamiltonian system with respect to $t$ compatible with $P_1^2$. A similar situation occurs for the $P_2^2$ 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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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 $H^{2+2+1}$
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 $H^{3+2}$ 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 $H^{2+1+1+1}$”, Ufa Math. J., 10:4 (2018), 92–102
V. A. Pavlenko, B. I. Suleimanov, ““Quantizations” of isomonodromic Hamilton system $H^{\frac{7}{2}+1}$”, 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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