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Instabilité des équations de Schrödinger

Abstract : In this work we study different instability phenomena for nonlinear Schrödinger equations.\\[3pt]
In the first part we show a phase decoherence mechanism for the semiclassical Gross-Pitaevski equation in dimension 3. This geometrical phenomenon occurs because the harmonical potential allows the construction of stationnary solutions to the equation which concentrate on circles of R^{3}.
In the second part, we obtain a geometric instability result for the cubic NLS on a riemannian surface. We assume that this surface admits a stable and nondegenerate periodic geodesic. Then with a WKB method we construct nonlinear quasimodes and we obtain approximate solutions to the equation for times such that instability occurs. Thus we generalize results of Burq-Gérard-Tzvetkov for the sphere.
In the last part, we consider supercritical Schrödinger equations on a riemannian manifold of dimension $d$. Thanks to nonlinear geometric optics in an analytic frame, we show a mechanism of loss of derivatives in Sobolev spaces, and an instabilty in the energy space.
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Contributor : Laurent Thomann Connect in order to contact the contributor
Submitted on : Tuesday, March 18, 2008 - 4:47:01 PM
Last modification on : Sunday, June 26, 2022 - 11:48:04 AM
Long-term archiving on: : Friday, September 28, 2012 - 11:21:45 AM


  • HAL Id : tel-00265284, version 1



Laurent Thomann. Instabilité des équations de Schrödinger. Mathématiques [math]. Université Paris Sud - Paris XI, 2007. Français. ⟨tel-00265284⟩



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