Editing Laplace's method (section)

==Further generalizations==
An extension of the ''steepest descent method'' is the so-called ''nonlinear stationary phase/steepest descent method''. Here, instead of integrals, one needs to evaluate asymptotically solutions of [[Riemann–Hilbert problem|Riemann–Hilbert factorization problem]]s.

Given a contour ''C'' in the [[complex sphere]], a function <math>f</math> defined on that contour and a special point, such as infinity, a holomorphic function ''M'' is sought  away from ''C'', with prescribed jump across ''C'', and with a given normalization at infinity. If <math>f</math> and hence ''M'' are matrices rather than scalars this is a problem that in general does not admit an explicit solution.

An asymptotic evaluation is then possible along the lines of the linear stationary phase/steepest descent method. The idea is to reduce asymptotically the solution of the given Riemann–Hilbert problem to that of a simpler, explicitly solvable, Riemann–Hilbert problem. Cauchy's theorem is used to justify deformations of the jump contour.

The nonlinear stationary phase was introduced by Deift and Zhou in 1993, based on earlier work of Its. A (properly speaking) nonlinear steepest descent method was introduced by Kamvissis, K. McLaughlin and P. Miller in 2003, based on previous work of Lax, Levermore, Deift, Venakides and Zhou. As in the linear case, "steepest descent contours" solve a min-max problem. In the nonlinear case they turn out to be "S-curves" (defined in a different context back in the 80s by Stahl, Gonchar and Rakhmanov).

The nonlinear stationary phase/steepest descent method has applications to the theory of soliton equations and [[integrable model]]s, [[random matrices]] and [[combinatorics]].