Asymptotics of Semiclassical Soliton Ensembles: Rigorous Justification of the WKB Approximation

Many important problems in the theory of integrable systems and approximation theory can be recast as Riemann-Hilbert problems for a matrix-valued unknown. Via the connection with approximation theory, and specifically the theory of orthogonal polynomials, one can also study problems from the theory of random matrix ensembles and combinatorics. Roughly speaking, solving a Riemann-Hilbert problem amounts to reconstructing a sectionally meromorphic matrix from given homogeneous multiplicative “jump conditions” at the boundary contours of the domains of meromorphy, from “principal part data” given at the prescribed singularities, and from a normalization condition. So, many asymptotic questions in integrable systems (e.g., long time behavior and singular perturbation theory) and approximation theory (e.g., behavior of orthogonal polynomials in the limit of large degree) amount to determining asymptotic properties of the solutionmatrix of a Riemann-Hilbert problem from given asymptotics of the jump conditions and principal part data. In recent years a collection of techniques has emerged for studying certain asymptotic problems of this sort. These techniques are analogous to familiar asymptotic methods for expanding oscillatory integrals, and we often refer to them as “steepestdescent” methods. The basic method first appeared in the work of Deift and Zhou [5]. The first applications were to Riemann-Hilbert problems without poles, in which the solution matrix is sectionally holomorphic. Later, some problems were studied in which