Riemann–Hilbert problem

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)).

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

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

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:

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).

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

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

In particular, Riemann–Hilbert factorization problems are used to extract asymptotic values 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 & Zhou (1993), expanding on a previous idea by Its (1982) and Manakov (1979). A crucial ingredient of the Deift–Zhou analysis is the asymptotic analysis of singular integrals on contours. The relevant kernel is the standard Cauchy kernel (see Gakhov (2001); also cf. the scalar example below).

An essential extension of the nonlinear method of stationary phase has been the introduction of the so-called finite gap g-function transformation by Deift, Venakides & Zhou (1997), which has been crucial in most applications. This was inspired by work of Lax, Levermore and Venakides, who reduced the analysis of the small dispersion limit of the KdV equation to the analysis of a maximization problem for a logarithmic potential under some external field: a variational problem of "electrostatic" type. The g-function is the logarithmic transform of the maximizing "equilibrium" measure. The analysis of the small dispersion limit of KdV equation has in fact provided the basis for the analysis of most of the work concerning "real" orthogonal polynomials (i.e. with the orthogonality condition defined on the real line) and Hermitian random matrices.

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. The corresponding variational problem is a max-min problem: one looks for a contour that minimizes the "equilibrium" measure. The study of the variational problem and the proof of existence of a regular solution, under some conditions on the external field, was done in Kamvissis & Rakhmanov (2005); the contour arising is an "S-curve", as defined and studied in the 1980s by Herbert R. Stahl, Andrei A. Gonchar and Evguenii A Rakhmanov.

An alternative asymptotic analysis of Riemann–Hilbert factorization problems is provided in McLaughlin & Miller (2006), especially convenient when jump matrices do not have analytic extensions. Their method is based on the analysis of d-bar problems, rather than the asymptotic analysis of singular integrals on contours. An alternative way of dealing with jump matrices with no analytic extensions was introduced in Varzugin (1996).

Another extension of the theory appears in Kamvissis & Teschl (2012) where the underlying space of the Riemann–Hilbert problem is a compact hyperelliptic Riemann surface. The correct factorization problem is no more holomorphic, but rather meromorphic, by reason of the Riemann–Roch theorem. The related singular kernel is not the usual Cauchy kernel, but rather a more general kernel involving meromorphic differentials defined naturally on the surface (see e.g. the appendix in Kamvissis & Teschl (2012)). The Riemann–Hilbert problem deformation theory is applied to the problem of stability of the infinite periodic Toda lattice under a "short range" perturbation (for example a perturbation of a finite number of particles).

Most Riemann–Hilbert factorization problems studied in the literature are 2-dimensional, i.e., the unknown matrices are of dimension 2. Higher-dimensional problems have been studied by Arno Kuijlaars and collaborators, see e.g. Kuijlaars & López (2015).

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

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

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

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.