Browsing by Author "Prokhorov, Andrei"

Now showing 1 - 4 of 4
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    Connection Problem for Painlevé Tau Functions
    (2019-08) Prokhorov, Andrei; Its, Alexander; Bleher, Pavel; Eremenko, Alexandre; Tarasov, Vitaly
    We derive the differential identities for isomonodromic tau functions, describing their monodromy dependence. For Painlev´e equations we obtain them from the relation of tau function to classical action which is a consequence of quasihomogeneity of corresponding Hamiltonians. We use these identities to solve the connection problem for generic solution of Painlev´e-III(D8) equation, and homogeneous Painlev´e-II equation. We formulate conjectures on Hamiltonian and symplectic structure of general isomonodromic deformations we obtained during our studies and check them for Painlev´e equations.
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    The Maxwell operator with periodic coefficients in a cylinder
    (2018) Filonov, N.; Prokhorov, Andrei; Mathematical Sciences, School of Science
    In the paper we consider the Maxwell operator in a three-dimensional cylinder with coefficients periodic along the axis of a cylinder. It is proved that for cylinders with circular and rectangular cross-section the spectrum of the Maxwell operator is absolutely continuous.
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    Monodromy dependence and connection formulae for isomonodromic tau functions
    (Duke, 2018) Its, Alexander R.; Lisovyy, O.; Prokhorov, Andrei; Mathematical Sciences, School of Science
    We discuss an extension of the Jimbo–Miwa–Ueno differential 1-form to a form closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational coefficients. This extension is based on the results of M. Bertola, generalizing a previous construction by B. Malgrange. We show how this 1-form can be used to solve a long-standing problem of evaluation of the connection formulae for the isomonodromic tau functions which would include an explicit computation of the relevant constant factors. We explain how this scheme works for Fuchsian systems and, in particular, calculate the connection constant for the generic Painlevé VI tau function. The result proves the conjectural formula for this constant proposed by Iorgov, Lisovyy, and Tykhyy. We also apply the method to non-Fuchsian systems and evaluate constant factors in the asymptotics of the Painlevé II tau function.
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    On the analysis of incomplete spectra in random matrix theory through an extension of the Jimbo–Miwa–Ueno differential
    (Elsevier, 2019-03) Bothner, Thomas; Its, Alexander; Prokhorov, Andrei; Mathematical Sciences, School of Science
    Several distribution functions in the classical unitarily invariant matrix ensembles are prime examples of isomonodromic tau functions as introduced by Jimbo, Miwa and Ueno (JMU) in the early 1980s [45]. Recent advances in the theory of tau functions [41], based on earlier works of B. Malgrange and M. Bertola, have allowed to extend the original Jimbo–Miwa–Ueno differential form to a 1-form closed on the full space of extended monodromy data of the underlying Lax pairs. This in turn has yielded a novel approach for the asymptotic evaluation of isomonodromic tau functions, including the exact computation of all relevant constant factors. We use this method to efficiently compute the tail asymptotics of soft-edge, hard-edge and bulk scaled distribution and gap functions in the complex Wishart ensemble, provided each eigenvalue particle has been removed independently with probability 1-γ ∈ (0, 1⌋.