On the Method of Steepest Descent

revised version: February 2005
Spyridon Kamvissis
(Please use for correspondence this email).

Submission date: 20. Feb. 2005
Pages: 28
Keywords and phrases: steepest descent, riemann-hilbert factorization
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Abstract:
We review the history of the nonlinear steepest descent method for the asymptotic evaluation of the solutions of Riemann-Hilbert factorization problems. We stress some recent results on the "non-self-adjoint" extension of the theory. In particular we consider the case of the semiclassical focusing NLS problem. We explain how the nonlinear steepest descent method gives rise to a maximin variational problem for Green potentials with external field in two dimensions and we announce results on existence and regularity of solutions to this variational problem.