Title

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

Authors

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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