Spyros Kamvisses: On Nonlinear Steepest Descent

The asymptotic analysis of integrable systems is often reducible to the asymptotic analysis of Riemann-Hilbert factorization problems. This is achieved through a Riemann-Hilbert deformation method, initiated by Its, and made systematic and rigorous by Deift and Zhou.
Although it is often known as the nonlinear steepest descent method, it is only recently that the term "steepest descent" has been justified and properly speaking steepest descent contours have been constructed. In fact, the term "steepest descent method" rather than, say, "stationary phase method" is only really meaningful in the case where the underlying Lax operator is non-self-adjoint. In my talk I will consider such a case, namely the case of the semiclassical focusing NLS problem. I will explain how the nonlinear steepest descent method gives rise to a maxi-min variational problem for Green potentials with external field in the complex plane and I will describe results on existence and regularity of solutions to this variational problem. The solutions are the steepest descent contours (S-curves; trajectories of quadratic
differentials) together with their equilibrium measures.

This is work in collaboration with K.McLaughlin, P.Miller, E.Rakhmanov.