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Ufimsk. Mat. Zh., 2018, Volume 10, Issue 4, Pages 92–102 (Mi ufa451)  

Solutions to analogues of non-stationary Schrödinger equations defined by isomonodromic Hamilton system $H^{2+1+1+1}$

V. A. Pavlenkoa, B. I. Suleimanovb

a Bashkir State Agrarian University, 50-letia Oktybray 34, 450001, Ufa, Russia
b Institute of Mathematics, Ufa Federal Research Center, RAS, Chernyshevskogo 112, 450008, Ufa, Russia

Abstract: We construct simultaneous solutions to two analogues of time-dependent solutions to Schrödinger equations defined by the Hamiltonians $H^{2+1+1+1}_{s_k}(s_1,s_2, q_1,q_2, p_1, p_2)$ $(k=1,2)$ to system $H^{2+1+1+1}$. This system is the first representative in a famous degenerations hierarchy of the Garnier system described in 1986 by H. Kimura. By an explicit symplectic transformation, this system reduces to a symmetric Hamilton system. In the constructions of this paper we rely mostly on linear systems of equations in the method of isomonodromic deformations for the system $H^{2+1+1+1}$ written out in 2012 in a paper by A. Kavakami, A. Nakamura and H. Sakai. These analogues of the non-stationary Schrödinger equations are evolution equations with times $s_1$ and $s_2$, which depend of two spatial variables. From the canonical non-stationary Schrödinger equations, these analogues arise as a result of the formal replacement of the Planck constant by $-2\pi i$. We construct the exact solutions to the two evolution equations in terms of the solutions to corresponding linear ordinary differential equations in the method of isomonodromic deformations. We discuss further prospects for constructing similar solutions to analogues of the non-stationary Schrödinger equations corresponding to the Hamiltonians of the entire degeneracy hierarchy of the Garnier system.

Keywords: Hamilton systems, Schrödinger equation, Painlevé equations, method of isomonodromic deformations.

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English version:
Ufa Mathematical Journal, 2018, 10:4, 92–102 (PDF, 388 kB); https://doi.org/10.13108/2018-10-4-92

Bibliographic databases:

UDC: 517.925
MSC: 34M56, 35Q41
Received: 01.08.2018

Citation: V. A. Pavlenko, B. I. Suleimanov, “Solutions to analogues of non-stationary Schrödinger equations defined by isomonodromic Hamilton system $H^{2+1+1+1}$”, Ufimsk. Mat. Zh., 10:4 (2018), 92–102; Ufa Math. J., 10:4 (2018), 92–102

Citation in format AMSBIB
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\by V.~A.~Pavlenko, B.~I.~Suleimanov
\paper Solutions to analogues of non-stationary Schr\"odinger equations defined by isomonodromic Hamilton system $H^{2+1+1+1}$
\jour Ufimsk. Mat. Zh.
\yr 2018
\vol 10
\issue 4
\pages 92--102
\mathnet{http://mi.mathnet.ru/ufa451}
\transl
\jour Ufa Math. J.
\yr 2018
\vol 10
\issue 4
\pages 92--102
\crossref{https://doi.org/10.13108/2018-10-4-92}
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