Abstract

The Benjamin-Ono equation is a model for internal water waves in stratified fluids over an infinitely-deep lower layer. It is similar to the Korteweg-de Vries (KdV) equation, but with a different dispersive term that is nonlocal, involving the Hilbert transform. Like KdV, it has a Lax pair, although its inverse-scattering transform is not fully justified in all details to date. This talk will describe the formation of dispersive shock waves in solutions of the Benjamin-Ono equation. Weak small-dispersion limits have been established for ``quasi’’ initial-value problems in the whole line setting by Miller and Xu and in the periodic setting by Gassot. After describing those results, we will discuss Whitham modulation theory due to Dobrokhotov and Krichever and explain some preliminary work on strong asymptotics to resolve the oscillations in the shock zone.