A nonlinear Fourier transform for the Benjamin–Ono equation on the torus and applications

Gérard, Patrick

Séminaire Laurent Schwartz — EDP et applications (2019-2020), Exposé no. 8, 19 p.

The Benjamin-Ono equation was introduced by Benjamin in 1967 as a model for a special regime of internal gravity waves at the interface of two fluids. This nonlinear dispersive equation admits a Lax pair structure involving nonlocal operators of Toeplitz type on the Hardy space. In the case of periodic boundary conditions, the spectral study of these Lax operators allows us to construct a nonlinear Fourier transform which conjugates the Benjamin–Ono dynamics to advection with constant velocity on tori. This construction has several applications: low regularity well-posedness of the initial value problem, long time behaviour of solutions and stability of traveling waves. This is a short report on these results, recently obtained in collaboration with T. Kappeler and P. Topalov.

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