Asymptotics of Orthogonal Polynomials with Complex Varying Quartic Weight: Global Structure, Critical Point Behavior and the First Painlev, Equation

    Authors

    M. Bertola;A. Tovbis

    Comments

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    Abbreviated Journal Title

    Constr. Approx.

    Keywords

    Asymptotics of orthogonal polynomials; Double scaling limit; Riemann; Hilbert problems; Painleve equations; ORDINARY DIFFERENTIAL-EQUATIONS; MONODROMY PRESERVING DEFORMATION; RATIONAL COEFFICIENTS; RESPECT; LIMIT; Mathematics

    Abstract

    We study the asymptotics of recurrence coefficients for monic orthogonal polynomials with the quartic exponential weight , where and . We consider in detail the points and , where the recurrence coefficients of the orthogonal polynomial exhibit a behavior that involves special solutions of the first Painlev, Riemann-Hilbert problem (RHP). Our principal concern is the description of their behavior in a neighborhood of special points in the -plane (accumulating at the indicated values) where the corresponding Painlev, function has poles. The nonlinear steepest descent method for the RHP is the main technique used in the paper. We note that the RHP near the critical points is very similar to the RHP describing the semiclassical limit of the focusing nonlinear Schrodinger equation near the point of gradient catastrophe that the present authors solved in 2013. Our approach is based on the technique developed in that earlier work. We also provide a numerical investigation of the "phase diagrams" in the -plane where the recurrence coefficients exhibit different asymptotic behaviors (nonlinear Stokes' phenomenon).

    Journal Title

    Constructive Approximation

    Volume

    41

    Issue/Number

    3

    Publication Date

    1-1-2015

    Document Type

    Article

    Language

    English

    First Page

    529

    Last Page

    587

    WOS Identifier

    WOS:000354472400008

    ISSN

    0176-4276

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