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.
@article{SLSEDP_2019-2020____A5_0,
author = {Patrick G\'erard},
title = {A nonlinear {Fourier} transform for {the~Benjamin{\textendash}Ono} equation on the torus and applications},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:8},
pages = {1--19},
publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2019-2020},
doi = {10.5802/slsedp.138},
zbl = {07436150},
language = {en},
url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.138/}
}
TY - JOUR AU - Patrick Gérard TI - A nonlinear Fourier transform for the Benjamin–Ono equation on the torus and applications JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:8 PY - 2019-2020 SP - 1 EP - 19 PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique UR - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.138/ DO - 10.5802/slsedp.138 LA - en ID - SLSEDP_2019-2020____A5_0 ER -
%0 Journal Article %A Patrick Gérard %T A nonlinear Fourier transform for the Benjamin–Ono equation on the torus and applications %J Séminaire Laurent Schwartz — EDP et applications %Z talk:8 %D 2019-2020 %P 1-19 %I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique %U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.138/ %R 10.5802/slsedp.138 %G en %F SLSEDP_2019-2020____A5_0
Patrick Gérard. A nonlinear Fourier transform for the Benjamin–Ono equation on the torus and applications. Séminaire Laurent Schwartz — EDP et applications (2019-2020), Talk no. 8, 19 p. doi : 10.5802/slsedp.138. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.138/
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