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pro vyhledávání: '"G��rard, Patrick"'
Autor:
G��rard, Patrick
We prove that the recently introduced spin Benjamin--Ono equation admits a Lax pair, and we deduce a family of conservation laws which allow to prove global wellposedness in all Sobolev spaces $H^k$ for every integer $k\geq 2$. We also infer an addit
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https://explore.openaire.eu/search/publication?articleId=doi_dedup___::1f21631d18ea653b02cf843d5786cd70
http://arxiv.org/abs/2202.08219
http://arxiv.org/abs/2202.08219
We construct inverse spectral theory for finite rank Hankel operators acting on the Hardy space of the upper half-plane. A particular feature of our theory is that we completely characterise the set of spectral data. As an application of this theory,
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https://explore.openaire.eu/search/publication?articleId=doi_________::0661bdbfb60cffb897bb976e829366d3
We prove smoothing properties of the solutions of the Benjamin-Ono equation in the Sobolev space $H^{s}(\mathbb{T},\mathbb{R})$ for any $s\ge 0$. To this end we show that Tao's gauge transform is a high frequency approximation of the nonlinear Fourie
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https://explore.openaire.eu/search/publication?articleId=doi_dedup___::965d2e759b11dde5d620b01b0c79d27d
http://arxiv.org/abs/2109.00610
http://arxiv.org/abs/2109.00610
In this paper, we survey our recent results on the Benjamin-Ono equation on the torus. As an application of the methods developed we construct large families of periodic or quasiperiodic solutions, which are not $C^\infty$-smooth.
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http://arxiv.org/abs/2103.09291
http://arxiv.org/abs/2103.09291
Autor:
G��rard, Patrick, Grellier, Sandrine
We derive an explicit formula for the general solution of the cubic Szeg�� equation and of the evolution equation of the corresponding hierarchy. As an application, we prove that all the solutions corresponding to finite rank Hankel operators are
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Autor:
G��rard, Patrick, M��hats, Florian
We study the Schr��dinger-Poisson system on the unit sphere $\SS^2$ of $\RR^3$, modeling the quantum transport of charged particles confined on a sphere by an external potential. Our first results concern the Cauchy problem for this system. We pr
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https://explore.openaire.eu/search/publication?articleId=doi_________::795bd848af2bbc9f37e160ed9de768eb
Autor:
G��rard, Patrick, Grellier, Sandrine
We consider the following Hamiltonian equation on the $L^2$ Hardy space on the circle, $$i\partial_tu=��(|u|^2u) ,$$ where $��$ is the Szeg�� projector. This equation can be seen as a toy model for totally non dispersive evolution equatio
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