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.
The classical problem, considered in Riemann's PhD dissertation, was that of finding a function analytic inside
is a circle, the problem reduces to deriving the Poisson formula.
[5] By the Riemann mapping theorem, it suffices to consider the case when
and so Hence the problem reduces to finding a pair of analytic functions
on the inside and outside, respectively, of the unit disk, so that on the unit circle and, moreover, so that the condition at infinity holds: Hilbert's generalization of the problem attempted to find a pair of analytic functions
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
(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.
near those points have to be posed to ensure uniqueness (see the scalar problem below).
Check: therefore, CAVEAT 1: If the problem is not scalar one cannot easily take logarithms.
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.
, the solution of the DBAR problem is[10] integrated over the entire complex plane; denoted by
For generalized analytic functions, this equation is replaced by in a region
[11] Generalized analytic functions have applications in differential geometry, in solving certain type of multidimensional nonlinear partial differential equations and multidimensional inverse scattering.
The so-called "nonlinear" method of stationary phase is due to Deift & Zhou (1993), expanding on a previous idea by Its (1982) and Manakov (1974) and using technical background results from Beals & Coifman (1984) and Zhou (1989).
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 (see Lax & Levermore (1983)).
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 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.
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 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.