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Modified scattering for the cubic Schrödinger equation on product spaces and applications
, 2013
"... Abstract. We consider the cubic nonlinear Schrödinger equation posed on the spatial domain R × Td. We prove modified scattering and construct modified wave operators for small initial and final data respectively (1 ≤ d ≤ 4). The key novelty comes from the fact that the modified asymptotic dynamics ..."
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Cited by 7 (3 self)
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Abstract. We consider the cubic nonlinear Schrödinger equation posed on the spatial domain R × Td. We prove modified scattering and construct modified wave operators for small initial and final data respectively (1 ≤ d ≤ 4). The key novelty comes from the fact that the modified asymptotic dynamics are dictated by the resonant system of this equation, which sustains interesting dynamics when d ≥ 2. As a consequence, we obtain global strong solutions (for d ≥ 2) with infinitely growing high Sobolev norms Hs. 1.
ASYMPTOTIC BEHAVIOR OF THE NONLINEAR SCHRÖDINGER EQUATION WITH HARMONIC TRAPPING
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On the continuous resonant equation for NLS: I. Deterministic analysis. Preprint: arXiv:1501.03760
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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Cited by 2 (2 self)
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
doi:10.1093/imrn/rnu230 Flow Monotonicity and Strichartz Inequalities
"... We identify complete monotonicity properties underlying a variety of wellknown sharp Strichartz inequalities in euclidean space. 1 ..."
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We identify complete monotonicity properties underlying a variety of wellknown sharp Strichartz inequalities in euclidean space. 1
ON DISCRETE RAREFACTION WAVES IN AN NLS TOY MODEL FOR WEAK TURBULENCE
"... We explore the rarefaction wavelike solutions to a model Hamiltonian dynamical system recently derived to study frequency cascades in the cubic defocusing nonlinear Schrödinger equation on the torus. ..."
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We explore the rarefaction wavelike solutions to a model Hamiltonian dynamical system recently derived to study frequency cascades in the cubic defocusing nonlinear Schrödinger equation on the torus.
Dedicated to Walter Craig on his 60th birthday TIMEAVERAGING FOR WEAKLY NONLINEAR CGL EQUATIONS WITH ARBITRARY POTENTIALS
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