In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane. Several existence theorems for Riemann–Hilbert problems have been produced by Mark Krein, Israel Gohberg and others (see the book by Clancey and Gohberg (1981)).

The Riemann problem

Suppose that ${\displaystyle \Sigma }$ is a closed simple contour in the complex plane dividing the plane into two parts denoted by ${\displaystyle \Sigma _{+}}$ (the inside) and ${\displaystyle \Sigma _{-}}$ (the outside), determined by the index of the contour with respect to a point. The classical problem, considered in Riemann's PhD dissertation (see Pandey (1996)), was that of finding a function

${\displaystyle M_{+}(z)=u(z)+iv(z)}$

analytic inside ${\displaystyle \Sigma _{+}}$ such that the boundary values of M₊ along ${\displaystyle \Sigma }$ satisfy the equation

${\displaystyle a(z)u(z)-b(z)v(z)=c(z)}$

for all ${\displaystyle z\in \Sigma }$, where a, b, and c are given real-valued functions (Bitsadze 2001).

By the Riemann mapping theorem, it suffices to consider the case when ${\displaystyle \Sigma }$ is the unit circle (Pandey 1996, §2.2). In this case, one may seek M₊(z) along with its Schwarz reflection:

${\displaystyle M_{-}(z)={\overline {M_{+}\left({\bar {z}}^{-1}\right)}}.}$

On the unit circle Σ, one has ${\displaystyle z=1/{\bar {z}}}$, and so

${\displaystyle M_{-}(z)={\overline {M_{+}(z)}},\quad z\in \Sigma .}$

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

${\displaystyle {\frac {a(z)+ib(z)}{2}}M_{+}(z)+{\frac {a(z)-ib(z)}{2}}M_{-}(z)=c(z),}$

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

${\displaystyle \lim _{z\to \infty }M_{-}(z)={\overline {{M}_{+}(0)}}.}$

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 ${\displaystyle \Sigma }$ one has

${\displaystyle \alpha (z)M_{+}(z)+\beta (z)M_{-}(z)=c(z)}$

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 ${\displaystyle \Sigma }$ 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 ${\displaystyle \Sigma }$. The Riemann–Hilbert problem is to find a pair of functions, M₊ and M₋ analytic, respectively, on the + and − side of ${\displaystyle \Sigma }$, subject to the equation

${\displaystyle \alpha (z)M_{+}(z)+\beta (z)M_{-}(z)=c(z)}$

for all z ∈ Σ.

Generalization: Matrix factorization problems

Given an oriented "contour" Σ (technically: an oriented union of smooth curves without points of infinite self-intersection 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:

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 outside Σ, 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. At end-points or intersection points of the contour Σ the jump condition is not defined; constraints on the growth of M near those points have to be posed to ensure uniqueness (see the scalar problem below).

Applications to integrability theory

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 problems for 1+1 dimensional partial differential equations on the line, or to periodic problems, or even to initial-boundary value problems (Fokas (2002)), can be stated as a Riemann–Hilbert problem. Likewise the inverse monodromy problem for Painlevé equations can be stated as a Riemann–Hilbert problem.

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 (Fokas, Its & Kitaev (1992)). Furthermore, the distribution of eigenvalues of random matrices in several classical ensembles is reduced to computations involving orthogonal polynomials (see for example Deift (2000)).

C. Combinatorial probability

The most celebrated example is the theorem of Baik, Deift & Johansson (1999) on the distribution of the length of the longest increasing subsequence of a random permutation. Together with the study of B above, it is one of the original rigorous investigations of so-called "integrable probability". But the connection between the theory of integrability and various classical ensembles of random matrices goes back to the work of Dyson (e.g.Dyson (1976)).

D. Connection to Donaldson-Thomas theory

The work of Bridgeland Bridgeland (2019) studies a class of Riemann-Hilbert problems coming from Donaldson-Thomas theory and makes connections with Gromov-Witten theory and exact WKB.

The numerical analysis of Riemann–Hilbert problems can provide an effective way for numerically solving integrable PDEs, see eg. Trogdon & Olver (2016).

Example: Scalar Riemann–Hilbert factorization problem

Suppose V = 2, and Σ is a contour from z = −1 to z = 1. Assuming M is bounded, what is the solution of M?

To solve this, let's take the logarithm of equation ${\displaystyle M_{+}=M_{-}V}$.

${\displaystyle \log M_{+}(z)=\log M_{-}(z)+\log 2.}$

Since M tends to 1, log M → 0 as z → ∞.

A standard fact about the Cauchy transform is that ${\displaystyle C_{+}-C_{-}=I}$ where ${\displaystyle C_{+},C_{-}}$ are the limits of the Cauchy transform from above and below Σ; therefore, we get

${\displaystyle {\frac {1}{2\pi i}}\int _{\Sigma _{+}}{\frac {\log 2}{\zeta -z}}\,d\zeta -{\frac {1}{2\pi i}}\int _{\Sigma _{-}}{\frac {\log {2}}{\zeta -z}}\,d\zeta =\log 2{\text{ when }}z\in \Sigma .}$

Because the solution M of a Riemann–Hilbert factorization problem is unique (an easy application of Liouville's theorem (complex analysis)), the Sokhotski–Plemelj theorem gives the solution. We get

${\displaystyle \log M={\frac {1}{2\pi i}}\int _{\Sigma }{\frac {\log {2}}{\zeta -z}}d\zeta ={\frac {\log 2}{2\pi i}}\int _{-1-z}^{1-z}{\frac {1}{\zeta }}d\zeta ={\frac {\log 2}{2\pi i}}\log {\frac {z-1}{z+1}},}$

i.e.

${\displaystyle M(z)=\left({\frac {z-1}{z+1}}\right)^{\frac {\log {2}}{2\pi i}}}$

which has a branch cut at contour ${\displaystyle \Sigma }$.

Check:

${\displaystyle {\begin{aligned}M_{+}(0)&=(e^{i\pi })^{\frac {\log 2}{2\pi i}}=e^{\frac {\log 2}{2}}\\M_{-}(0)&=(e^{-i\pi })^{\frac {\log 2}{2\pi i}}=e^{-{\frac {\log 2}{2}}}\end{aligned}}}$

therefore,

${\displaystyle M_{+}(0)=M_{-}(0)e^{\log {2}}=M_{-}(0)2.}$

CAVEAT 1: If the problem is not scalar one cannot easily take logarithms. In general explicit solutions are very rare.

CAVEAT 2: The boundedness (or at least a constraint on the blow-up) of M near the special points 1 and -1 is crucial. Otherwise any function of the form

${\displaystyle M(z)=\left({\frac {z-1}{z+1}}\right)^{\frac {\log {2}}{2\pi i}}+{\frac {a}{z-1}}+{\frac {b}{z+1}}}$

is also a solution. In general, conditions on growth are necessary at special points (the end-points of the jump contour or intersection point) to ensure that the problem is well-posed.

See also

• Riemann–Hilbert correspondence

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