# Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds

Sun, Ruoci

##### Abstract:
This paper is dedicated to proving the complete integrability of the Benjamin-Ono (BO) equation on the line when restricted to every N-soliton manifold, denoted by U-N. We construct generalized action-angle coordinates which establish a real analytic symplectomorphism from U-N onto some open convex subset of R-2N and allow to solve the equation by quadrature for any such initial datum. As a consequence, U-N is the universal covering of the manifold of N-gap potentials for the BO equation on the torus as described by Gerard-Kappeler (Commun Pure Appl Math, 2020. https://doioorg/10 . 1002/cpa.21896. arXiv:1905.01849). The global well-posedness of the BO equation on U-N is given by a polynomial characterization and a spectral characterization of the manifold U-N. 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.

 Zugehörige Institution(en) am KIT Institut für Analysis (IANA) Publikationstyp Zeitschriftenaufsatz Publikationsjahr 2021 Sprache Englisch Identifikator ISSN: 0010-3616, 1432-0916 KITopen-ID: 1000131160 Erschienen in Communications in Mathematical Physics Verlag Springer Band 383 Seiten 1051–1092 Vorab online veröffentlicht am 26.03.2021 Nachgewiesen in ScopusWeb of Science
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