The ∂ Steepest Descent Method and the Asymptotic Behavior of Polynomials Orthogonal on the Unit Circle with Fixed and Exponentially Varying Nonanalytic Weights

By K. Mclaughlin, Peter D. Miller. 2004

165 citations recorded by the source corpus.

The steepest descent method for asymptotic analysis of matrix Riemann-Hilbert problems was introduced by Deift and Zhou in 1993 [14]. A matrix Riemann-Hilbert problem is specified by giving a triple (Σ, v,N) consisting of an oriented contour Σ in the complex z-plane, a matrix function v : Σ → SL(N) which is usually taken to be continuous except at self-intersection points of Σwhere a certain compatibility condition is required, and a normalization condition N as z → ∞. If Σ is not bounded, certain asymptotic conditions are required of v in order to have compatibility with the normalization condition. Consider an analytic functionM : C \ Σ → SL(N) taking continuous boundary valuesM+(z) (resp., M−(z)) on Σ from the left (resp., right). The Riemann-Hilbert problem (Σ, v,N) is then to find such a matrix M(z) satisfying the normalization condition N as z → ∞ and the jump condition M+(z) = M−(z)v(z) whenever z is a non-self-intersection point of Σ (so the left and right boundary values are indeed well defined). The steepest descent method of Deift and Zhou applies to certain Riemann-Hilbert problems where the jumpmatrix v(z) depends on an auxiliary control parameter, and is a method for extracting asymptotic properties of the solution M(z) (and indeed proving the existence and

Subfields: Applied Mathematics, Computational Theory and Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c63fe3aed89f391b3b04a856f04bda25ea39ce40

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Citations in the corpus

Universality limits in the bulk for varying measures (2008, methodology): “[3] J. Baik, T.Kriecherbauer, K.McLaughlin and P.Miller, Uniform asymptotics for poly- nomials orthogonal with respect to a general class of discrete weights and universality results for associated Ensembles, Annals of Math.”
Long time asymptotic behavior of the focusing nonlinear Schrödinger equation (2016, unknown): “Recently, McLaughlin and Miller [15, 16], developed a method of asymptotic analysis of Riemann-Hilbert problems based on ∂ problems, rather than the asymptotic analysis of singular integrals on contours.”
Szego Orthogonal Polynomials with Respect to an Analytic Weight: Canonical Representation and Strong Asymptotics (2005, methodology): “The m ain technique isthe steepestdescentanalysisofDeiftand Zhou,based on them atrix Riem ann-Hilbert characterization proposed by Fokas,Itsand K itaev.”
Long-Time Asymptotics of a Three-Component Coupled mKdV System (2019, background): “McLaughlin and Miller [19] generalized the steepest descent method to the case when the jump matrix fails to be analytic and their method is now called a nonlinear ∂̄ steepest descent method.”
On the Asymptotic Stability of N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N}$$\end{document}-Soliton Solutions (2016, methodology): “Our proof uses the ∂ method for contour deformation introduced in McLaughlin–Miller [37,38] and Dieng–McLaughlin [21], which allows us to consider initial data with only a small amount of regularity while simultaneously simplifying many of the necessary estimates of more standard steepest descent.”
The Steepest Descent Method for Orthogonal Polynomials on the Real Line with Varying Weights (2008, methodology): “With the exception of [12] and [15], all of these applications and extensions deal with weights that are real analytic.”
Critical Measures, Quadratic Differentials, and Weak Limits of Zeros of Stieltjes Polynomials (2009, methodology): “The recently developed tools of asymptotic analysis of the RH problems of a certain class, such as the non-linear steepest descent method of Deift and Zhou (see [22], [23], [24], as well as [10]) combined with the ∂-problem ([62], [63]), have become powerful weapons in the study of the strong asympt”
Long-Time Asymptotics for the Korteweg–de Vries Equation via Nonlinear Steepest Descent (2008, unknown): “For a general result which applies in the case where R(k) has modulus strictly less than one and no solitons are present we refer to Varzugin [35] and for another recent generalization of the nonlinear steepest descent method to McLaughlin and Miller [28].”