Seminar Event Detail


Differential Equations

Date:  Thursday, September 30, 2021
Location:  umich.zoom.us/j/98365676067 Virtual (4:00 PM to 5:00 PM)

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

Abstract:   Zoom: https://umich.zoom.us/j/98365676067

Abstract: This presentation, which is based on the work Sun [2], 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 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 [1]. 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. The construction of action-angle coordinates for each U_N constitutes a first step towards the soliton resolution conjecture of the BO equation on the line.

[1] Gerard, P., Kappeler, T. On the integrability of the Benjamin-Ono equation on the torus, Commun. Pure Appl. Math. 74 (2021), no.8, 1685-1747, https://doi.org/10.1002/cpa.21896, 2021.
[2] Sun, R. Complete integrability of the Benjamin-Ono equation on the multi-soliton manifolds, Commun. Math. Phys. 383, 1051-1092 (2021). https://doi.org/10.1007/s00220-021-03996-1

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Speaker:  Ruoci Sun
Institution:  Karlsruhe Institute of Technology

Event Organizer:     

 

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