Algebra and Geometry Seminar: Ruoci Sun
Event Description:
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.