Riemann-Hilbert

: "For the original problem of Hilbert concerning the existence of linear differential equations having a given monodromy group see Hilbert's twenty-first problem."

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)).

The Riemann problem

Suppose that Σ is a closed simple contour in the complex plane dividing the plane into two parts denoted by Σ₊ (the inside) and Σ₋ (the outside), determined by the index of the contour with respect to a point. The classical problem, considered in Riemann's PhD dissertation (see harvtxt|Pandey|1996), was that of finding a function

:$M\_+(z)\; =\; u(z)\; +\; i\; v(z)!$

analytic inside Σ such that the boundary values of "M"₊ along Σ satisfy the equation

:$a(z)u(z)\; -\; b(z)v(z)\; =\; c(z)\; !$

for all "z"∈Σ, where "a", "b", and "c" are given real-valued functions harv|Bitsadze|2001.

By the Riemann mapping theorem, it suffices to consider the case when Σ is the unit circle harv|Pandey|1996|loc=§2.2. In this case, one may seek "M"₊("z") along with its Schwarz reflection:

:

On the unit circle Σ, one has , and so

:$M\_-(z)\; =\; overline\{M\_+(z)\},quad\; zinSigma.$

Hence the problem reduces to finding a pair of functions "M"₊("z") and "M"_-("z") analytic, respectively, on the inside and the outside of the unit disc, so that on the unit circle

:$frac\{a(z)+ib(z)\}\{2\}M\_+(z)\; +\; frac\{a(z)-ib(z)\}\{2\}M\_-(z)\; =\; c(z),$

and, moreover, so that the condition at infinity holds:

:

The Hilbert problem

Hilbert's generalization was to consider the problem of attempting to find "M"₊ and "M"_- analytic, respectively, on the inside and outside of the curve Σ, such that on Σ one has

:

where α, β, and "c" are arbitrary given complex-valued functions (no longer just complex conjugates).

Riemann-Hilbert problems

In the Riemann problem as well as Hilbert's generalization, the contour Σ was simple. A full Riemann-Hilbert problem allows that the contour may be composed of a union of several oriented smooth curves, with no intersections. The + and − sides of the "contour" may then be determined according to the index of a point with respect to Σ. The Riemann-Hilbert problem is to finding a pair of functions, "M"₊ and "M"_- analytic, respectively, on the + and − side of Σ, subject to the equation

:

for all "z"∈Σ.

Generalization: factorization problems

Given an oriented "contour" Σ (now meaning an oriented union of smooth curves without self-intersections in the complex plane). 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:
# 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.
#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 "L"_"2" 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 harvtxt|Deift|1999).

C. Combinatorial probability. The most celebrated example is the theorem of harvtxt|Baik|Deift|Johansson|1999 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 harvtxt|Deift|Zhou|1993, expanding on a previous idea by harvtxt|Its|1982.

An essential extension of the nonlinear method of stationary phase has been the introduction of the so-called g-function transformation by harvtxt|Deift|Venakides|Zhou|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 harvtxt|Kamvissis|McLaughlin|Miller|2003. In that case, actual "steepest descent contours" are defined and computed.

References

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