Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds
November 9, 2021
This presentation is dedicated to describing the complete integrability of the Benjamin–Ono (BO) equation on the line when restricted to every N-soliton manifold, denoted by UN . We construct (generalized) action–angle coordinates which establish a real analytic symplectomorphism from UN onto some open convex subset of R^2N 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 characterization 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 infinitesimal generator of the very shift semigroup. The construction of action–angle coordinates for each UN constitutes a first
step towards the soliton resolution conjecture of the BO equation on the line.