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Rational normal forms and stability of small solutions to nonlinear Schrà ¶dinger equations Bernier, Joackim
Description
Considering general classes of nonlinear Schr\"odinger equations on the circle with nontrivial cubic part and without external parameters, I will present the construction a new type of normal forms, namely rational normal forms, on open sets surrounding the origin in high Sobolev regularity. With this new tool we prove that, given a large constant $M$ and a sufficiently small parameter $\varepsilon$, for generic initial data of size $\varepsilon$, the flow is conjugated to an integrable flow up to an arbitrary small remainder of order $\varepsilon^{M+1}$. This implies that for such initial data $u(0)$ we control the Sobolev norm of the solution $u(t)$ for time of order $\varepsilon^{-M}$. Furthermore this property is locally stable: if $v(0)$ is sufficiently close to $u(0)$ (of order $\varepsilon^{3/2}$) then the solution $v(t)$ is also controled for time of order $\varepsilon^{-M}$. This is a joint work with Erwan Faou and Benoà ®t Grà ©bert.
Item Metadata
Title |
Rational normal forms and stability of small solutions to nonlinear Schrà ¶dinger equations
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2019-06-10T12:17
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Description |
Considering general classes of nonlinear Schr\"odinger equations on the circle with nontrivial cubic part and without external parameters, I will present the construction a new type of normal forms, namely rational normal forms, on open sets surrounding the origin in high Sobolev regularity. With this new tool we prove that, given a large constant $M$ and a sufficiently small parameter $\varepsilon$, for generic initial data of size $\varepsilon$, the flow is conjugated to an integrable flow up to an arbitrary small remainder of order $\varepsilon^{M+1}$. This implies that for such initial data $u(0)$ we control the Sobolev norm of the solution $u(t)$ for time of order $\varepsilon^{-M}$. Furthermore this property is locally stable: if $v(0)$ is sufficiently close to $u(0)$ (of order $\varepsilon^{3/2}$) then the solution $v(t)$ is also controled for time of order $\varepsilon^{-M}$. This is a joint work with Erwan Faou and Benoà ®t Grà ©bert.
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Extent |
43.0 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: Institut de recherche mathématique de Rennes
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Series | |
Date Available |
2019-12-08
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0386787
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Graduate
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International