TITLE:  Complete Integrability of the Benjamin-Ono Equation

 

ABSTRACT: The Benjamin-Ono equation (BO) is a model for deep gravity waves in stratified fluids. It was discovered to be completely integrable by Nakamura, Bock and Kruskal, and Fokas and Ablowitz. Unlike many other completely integrable equations, the direct and inverse scattering transform (IST) for BO is only established on a formal level. In this talk, we will discuss several results aiming at a rigorous analysis of the foundation of the IST for BO. Such a problem will require spectral analysis of a certain singular integral perturbation of the derivative operator, together with a good understanding of a non-local Riemann-Hilbert problem.