Doubly periodic solutions of the Benjamin-Ono equation (Video https://www.youtube.com/watch?v=P2Smd2y9nvs)

We investigate the existence of non trivial doubly periodic solutions of the Benjamin-Ono equation, depending on the space period, the time period, the space average, and the space regularity of the solution. In particular, for doubly periodic solutions with space average 0 and with time period and space period equal to 2\pi, we prove that L^2 regularity implies analyticity, while there exists more such solutions which belong to the Sobolev space H^s for every s<0. The crucial tool is the Birkhoff map for the Benjamin-Ono equation recently obtained in collaboration with T. Kappeler and P. Topalov.