I will define Riemann-Hilbert problems and I will show how, in some simple cases, problems that arise form initial value problems for soliton equations can be symptotically reduced to explicitly solvable ones. Also I will explain the nonlinear stationary phase idea of A.its.
Spyridon Kamvissis received his Ph.D. from Courant Institute New York University with Peter Lax and Percy Deift, and a Habilitation from University of Paris VII (Jussieu).
His research has focused mostly on "completely integrable" infinite dimensional Hamiltonian systems, like the KdV equation, the nonlinear Schrödinger equation, and the Toda lattice. Particularly interested in asymptotic problems like the investigation of long time asymptotics, semiclassical asymptotics, zero dispersion limits and continuum limits of solutions of initial and initial-boundary value problems for nonlinear dispersive partial differential equations and nonlinear lattices, including difficult problems involving instabilities (like the so-called modulational instability). He has used and extended techniques from PDE theory, complex analysis, harmonic analysis, potential theory and algebraic geometry. Along the way, has made contributions to the analysis of Riemann-Hilbert factorization problems on the complex plane or a hyperelliptic Riemann surface and the theory of variational problems for Green potentials with external harmonic fields. In a sense, he has worked on a "nonlinear microlocal analysis" that generalizes the classical theory of stationary phase and steepest descent.