**Abstract:** |
Asymptotic models for internal wave motion in 1+1 dimensions include nonlocal linear dispersion terms arising from the elimination of potential flow on one side of the interface via a Dirichlet-Neumann map. Such models include the intermediate long wave equation in the case of finite depth and the Benjamin-Ono equation in the case of infinite depth (of the lower fluid layer). In some situations it is physically reasonable to assume that the dispersive effects are formally small compared with nonlinear effects that eventually lead to wave breaking, and then it is interesting to study the effect that weak dispersion has as a regularizing effect on the breaking waves. This problem has been studied for many years in the context of the Korteweg-de Vries equation, with key ideas going back to Whitham, Gurevich-Pitaevskii, and Lax-Levermore, and with more modern developments such as the results of Claeys and Grava arising from the Deift-Zhou steepest descent method for Riemann-Hilbert problems. In this talk, I will describe some of our attempts to study the corresponding problem in the context of the Benjamin-Ono equation. In particular, we will present a simple and intuitive weak convergence result (joint work with Z. Xu) that is a consequence of a new analogue of the variational method of Lax and Levermore but that takes as inspiration also the moment expansion method of Wigner in random matrix theory. We will also describe more recent results on the direct scattering problem for rational potentials (joint work with A. Wetzel). |