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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.

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Verlagsausgabe §
DOI: 10.5445/IR/1000131160
Veröffentlicht am 07.04.2021
Cover der Publikation
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 Scopus
Web of Science
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