Riemann-Hilbert

In mathematics, Riemann-Hilbert problems are a class of problems that arise, inter alia, in the study of differential equations in the complex plane. Several existence theorems for Riemann-Hilbert problems have been produced by Krein, Gohberg and others (see the book by Clancey and Gohberg (1981)).

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Formulation

Given an oriented "contour" Σ (meaning an oriented union of smooth curves in the complex plane), we distinguish between the + and - sides using the convention that + is to the left and - is to the right. A Riemann-Hilbert factorization problem is the following.

Given a matrix function V defined on the contour Σ, to find a holomorphic matrix function M defined on the complement of Σ, such that two conditions be satisfied:

1. If M₊ and M_- denote the non-tangential limits of M as we approach Σ, then M₊=M_-V, at all points of non-intersection in Σ.
2. As z tends to infinity along any direction, M tends to the identity matrix.

In the simplest case V is smooth and integrable. In more complicated cases it could have singularities. The limits M₊ and M_- could be classical and continuous or they could be taken in the L₂ sense.

Applications

Riemann-Hilbert problems have applications to several related classes of problems.

A. Integrable models. The inverse scattering or inverse spectral problem associated to the Cauchy problem for 1+1 dimensional partial differential equations on the line, periodic problems, or even initial-boundary value problems, can be stated as Riemann-Hilbert problems.

B. Orthogonal polynomials, Random matrices. Given a weight on a contour, the corresponding orthogonal polynomials can be computed via the solution of a Riemann-Hilbert factorization problem. Furthermore, the distribution of eigenvalues of random matrices in several ensembles is reduced to computations involving orthogonal polynomials (see for example Deift (1999)).

C. Combinatorial probability. The most celebrated example is the theorem of Baik, Deift and Johansson on the distribution of the length of the longest increasing subsequence of a random permutation.

In particular, Riemann-Hilbert factorization problems are used to extract asymptotics for the three problems above (say, as time goes to infinity, or as the dispersion coefficient goes to zero, or as the polynomial degree goes to infinity, or as the size of the permutation goes to infinity). There exists a method for extracting the asymptotic behavior of solutions of Riemann-Hilbert problems, analogous to the method of stationary phase and the method of steepest descent applicable to exponential integrals.

By analogy with the classical asymptotic methods, one "deforms" Riemann-Hilbert problems which are not explicitly solvable to problems that are. The so-called "nonlinear" method of stationary phase is due to Deift and Zhou (1993), expanding on a previous idea by Alexander Its (1982).

An essential extension of the nonlinear method of stationary phase has been the introduction of the so-called g-function transformation by Deift, Venakides, Zhou in 1997, inspired by previous results of Lax and Levermore, and has been crucial in most applications. Perhaps the most sophisticated extension of the theory so far is the one applied to the "non self-adjoint" case, i.e. when the underlying Lax operator (the first component of the Lax pair) is not self-adjoint, by Kamvissis, McLaughlin, Miller (2003). In that case, actual "steepest descent contours" are defined and computed.

References

• J. Baik, P. Deift, K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, Journal of AMS, 12, 1999, 1119-1178. [1]

• K. Clancey, I. Gohberg, Factorization of matrix functions and singular integral operators, Oper. Theory: Advances and Appl., 3, Birkhäuser Verlag, Basel-Boston-Stuttgart, 1981.

• P. A. Deift, "Orthogonal Polynomials and Random Matrices", AMS, 1999.

• P. Deift, S. Venakides, and X. Zhou, New Results in Small Dispersion KdV by an Extension of the Steepest Descent Method for Riemann-Hilbert Problems, International Mathematical Research Notices 1997, 286-299.

• P. Deift, X. Zhou, A Steepest Descent Method for Oscillatory Riemann-Hilbert Problems; Asymptotics for the MKdV Equation, Ann. of Math. (2), v.137 (1993), no. 2, 295–368.

• A.R.Its, Asymptotics of Solutions of the Nonlinear Schrödinger Equation and Isomonodromic Deformations of Systems of Linear Differential Equations, Soviet Mathematics Doklady, v.24, n.3, 1982, 14--18.

• A.R.Its, The Riemann-Hilbert Problem and Integrable Systems, Notices of the AMS, 50-11 (2003), 1389--1400. [2]

• S. Kamvissis, K. McLaughlin, P.Miller, Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation, Annals of Mathematics, Study 154, Princeton University Press, Princeton, 2003.

• P.D.Lax, C.D.Levermore, The Zero Dispersion Limit for the KdV Equation, I-III, Communications on Pure and Applied Mathematics, v.36, 1983, 253-290, 571-593, 809-829.