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TMF, 2008, Volume 156, Number 3, Pages 364–377 (Mi tmf6253)  

This article is cited in 14 scientific papers (total in 14 papers)

“Quantizations” of the second Painlevé equation and the problem of the equivalence of its $L$$A$ pairs

B. I. Suleimanov

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

Abstract: We show that the known Flaschka–Newell $L$$A$ pair for the second Painlevé equation gives solutions to linear evolutionary equations similar to the quantum Schrödinger equations. Using the Fourier transform for distributions, we derive this pair from the classical Garnier pair.

Keywords: quantum Schrödinger equation, Hamiltonian, Painlevé equations, isomonodromic deformation


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English version:
Theoretical and Mathematical Physics, 2008, 156:3, 1280–1291

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Received: 23.05.2007
Revised: 09.01.2008

Citation: B. I. Suleimanov, ““Quantizations” of the second Painlevé equation and the problem of the equivalence of its $L$$A$ pairs”, TMF, 156:3 (2008), 364–377; Theoret. and Math. Phys., 156:3 (2008), 1280–1291

Citation in format AMSBIB
\by B.~I.~Suleimanov
\paper ``Quantizations'' of the~second Painlev\'e equation and the~problem of
the~equivalence of its $L$--$A$ pairs
\jour TMF
\yr 2008
\vol 156
\issue 3
\pages 364--377
\jour Theoret. and Math. Phys.
\yr 2008
\vol 156
\issue 3
\pages 1280--1291

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    This publication is cited in the following articles:
    1. D. P. Novikov, “The $2{\times}2$ matrix Schlesinger system and the Belavin–Polyakov–Zamolodchikov system”, Theoret. and Math. Phys., 161:2 (2009), 1485–1496  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. V. V. Tsegel'nik, “Hamiltonians associated with the third and fifth Painlevé equations”, Theoret. and Math. Phys., 162:1 (2010), 57–62  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. B. I. Suleimanov, ““Kvantovaya” linearizatsiya uravnenii Penleve kak komponenta ikh $L,A$ par”, Ufimsk. matem. zhurn., 4:2 (2012), 127–135  mathnet
    4. Zabrodin A. Zotov A., “Quantum Painlevé-Calogero Correspondence”, J. Math. Phys., 53:7 (2012), 073507  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    5. A. V. Zotov, A. V. Smirnov, “Modifications of bundles, elliptic integrable systems, and related problems”, Theoret. and Math. Phys., 177:1 (2013), 1281–1338  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. B. I. Suleimanov, ““Quantizations” of Higher Hamiltonian Analogues of the Painlevé I and Painlevé II Equations with Two Degrees of Freedom”, Funct. Anal. Appl., 48:3 (2014), 198–207  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    7. Levin A. Olshanetsky M. Zotov A., “Planck Constant as Spectral Parameter in Integrable Systems and Kzb Equations”, J. High Energy Phys., 2014, no. 10, 109  crossref  mathscinet  zmath  isi  scopus  scopus
    8. Kartak V.V., ““Painlevé 34” Equation: Equivalence Test”, Commun. Nonlinear Sci. Numer. Simul., 19:9 (2014), 2993–3000  crossref  mathscinet  adsnasa  isi  scopus  scopus
    9. Rumanov I., “Beta Ensembles, Quantum Painlevé Equations and Isomonodromy Systems”, Algebraic and Analytic Aspects of Integrable Systems and Painlev? Equations, Contemporary Mathematics, 651, ed. Dzhamay A. Maruno K. Ormerod C., Amer Mathematical Soc, 2015, 125–155  crossref  mathscinet  zmath  isi
    10. 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
    11. 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
    12. Tamara Grava, Alexander Its, Andrei Kapaev, Francesco Mezzadri, “On the Tracy–Widom$_\beta$ Distribution for $\beta=6$”, SIGMA, 12 (2016), 105, 26 pp.  mathnet  crossref
    13. 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
    14. 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
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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