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“Quantizations” of higher Hamiltonian analogues of the Painlevé I and Painlevé II equations with two degrees of freedom

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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 21 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 21 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 21 . A similar situation occurs for the P 22 equation in the Painlevé II hierarchy.

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Authors and Affiliations

  1. Institute of Mathematics and Computer Center, Ufa Scientific Center, Russian Academy of Sciences, Ufa, Russia

    B. I. Suleimanov

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__________

Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 48, No. 3, pp. 52–62, 2014

Original Russian Text Copyright © by B. I. Suleimanov

This work was supported by the Russian Science Foundation, grant 14-11-00078.

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Suleimanov, B.I. “Quantizations” of higher Hamiltonian analogues of the Painlevé I and Painlevé II equations with two degrees of freedom. Funct Anal Its Appl 48, 198–207 (2014). https://doi.org/10.1007/s10688-014-0061-0

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  • DOI: https://doi.org/10.1007/s10688-014-0061-0

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