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RiemannHilbert
In mathematics, RiemannHilbert problems are a class of problems that arise, inter alia, in the study of differential equations in the complex plane. Several existence theorems for RiemannHilbert problems have been produced by Krein, Gohberg and others (see the book by Clancey and Gohberg (1981)). Additional recommended knowledgeFormulationGiven 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 RiemannHilbert 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 L_{2} sense. ApplicationsRiemannHilbert 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 initialboundary value problems, can be stated as RiemannHilbert problems. B. Orthogonal polynomials, Random matrices. Given a weight on a contour, the corresponding orthogonal polynomials can be computed via the solution of a RiemannHilbert 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, RiemannHilbert 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 RiemannHilbert 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" RiemannHilbert problems which are not explicitly solvable to problems that are. The socalled "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 socalled gfunction 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 selfadjoint" case, i.e. when the underlying Lax operator (the first component of the Lax pair) is not selfadjoint, by Kamvissis, McLaughlin, Miller (2003). In that case, actual "steepest descent contours" are defined and computed. References

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "RiemannHilbert". A list of authors is available in Wikipedia. 