Zero-dispersion limit for the Benjamin-Ono equation

We consider the Benjamin-Ono equation on the line with a small dispersion parameter going to zero. After the shock time for the underlying inviscid Burgers equation, a dispersive shock wave appears in the solution when the parameter is small enough. We show that the solution is asymptotic to the multi-phase solution of Dobrokhotov and Krichever (generalizing periodic traveling waves) for the Benjamin-Ono equation, modulated by slow-varying parameters that depend only on the branches of the Burgers equations obtained by the method of characteristics. The proof relies on a solution formula of the Benjamin-Ono equation established by Gérard [28] and that we simplify for rational initial data in [7]. A paper on the zero-dispersion asymptotics will appear soon in [8].