Patrick Gérard: Long-time dynamics for the Benjamin-Ono equation on the line

Introduced in 1967 as a model of fluid mechanics, the Benjamin-Ono equation turns out to be a striking example of an infinite-dimensional integrable system. I will review its main properties, including a Lax pair and an explicit representation formula for its solutions, and I will try to explain how they have been recently used to prove a conjecture on the role of soliton solutions in the long-time dynamics on the line.