Title: Complete integrability of the Benjamin--Ono equation on the multi-soliton manifolds

Abstract: This presentation, which is based on the work Sun , is dedicated to describing the complete

integrability of the Benjamin{Ono (BO) equation on the line when restricted to every N-soliton mani-

fold, denoted by UN. We construct (generalized) action{angle coordinates which establish a real analytic

symplectomorphism from UN onto some open convex subset of R2N and allow to solve the equation by

quadrature for any such initial datum. As a consequence, UN is the universal covering of the manifold

of N-gap potentials for the BO equation on the torus as described by Gerard Kappeler. The global

well-posedness of the BO equation on UN is given by a polynomial characterization and a spectral char-

acterization of the manifold UN. Besides the spectral analysis of the Lax operator of the BO equation

and the shift semigroup acting on some Hardy spaces, the construction of such coordinates also relies on

the use of a generating functional, which encodes the entire BO hierarchy. The inverse spectral formula

of an N-soliton provides a spectral connection between the Lax operator and the in nitesimal generator

of the very shift semigroup. The construction of action{angle coordinates for each UN constitutes a  rst

step towards the soliton resolution conjecture of the BO equation on the line.