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Funktsional. Anal. i Prilozhen., 2014, Volume 48, Issue 3, Pages 52–62 (Mi faa3150)  

This article is cited in 8 scientific papers (total in 8 papers)

“Quantizations” of Higher Hamiltonian Analogues of the Painlevé I and Painlevé II Equations with Two Degrees of Freedom

B. I. Suleimanov

Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences, Ufa

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.

Keywords: quantization, Schrödinger equation, Hamiltonian, Painlevé equations, isomonodromic deformations, integrability

Funding Agency Grant Number
Russian Science Foundation 14-11-00078


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English version:
Functional Analysis and Its Applications, 2014, 48:3, 198–207

Bibliographic databases:

UDC: 517.9
Received: 18.04.2012

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

Citation in format AMSBIB
\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
\jour Funct. Anal. Appl.
\yr 2014
\vol 48
\issue 3
\pages 198--207

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    This publication is cited in the following articles:
    1. 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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    2. 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  mathnet  crossref  mathscinet  isi  elib
    3. V. A. Pavlenko, B. I. Suleimanov, ““Quantizations” of isomonodromic Hamilton system $H^{\frac{7}{2}+1}$”, Ufa Math. J., 9:4 (2017), 97–107  mathnet  crossref  isi  elib
    4. V. I. Kachalov, Yu. S. Fedorov, “O metode malogo parametra v nelineinoi matematicheskoi fizike”, Sib. elektron. matem. izv., 15 (2018), 1680–1686  mathnet  crossref
    5. 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  mathnet  crossref  isi
    6. Adler V.E., “Nonautonomous Symmetries of the Kdv Equation and Step-Like Solutions”, J. Nonlinear Math. Phys., 27:3 (2020), 478–493  crossref  mathscinet  zmath  isi
    7. 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  mathnet  crossref  crossref  mathscinet  isi
    8. 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  mathnet  crossref  isi
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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