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THE WEAKLY NONLINEAR LARGE BOX LIMIT OF THE 2D CUBIC NONLINEAR SCHRÖDINGER EQUATION
, 2013
"... We consider the cubic nonlinear Schrödinger (NLS) equation set on a two dimensional box of size L with periodic boundary conditions. By taking the large box limit L→ ∞ in the weakly nonlinear regime (characterized by smallness in the critical space), we derive a new equation set on R2 that approx ..."
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Cited by 12 (4 self)
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We consider the cubic nonlinear Schrödinger (NLS) equation set on a two dimensional box of size L with periodic boundary conditions. By taking the large box limit L→ ∞ in the weakly nonlinear regime (characterized by smallness in the critical space), we derive a new equation set on R2 that approximates the dynamics of the frequency modes. This nonlinear equation turns out to be Hamiltonian and enjoys interesting symmetries, such as its invariance under the Fourier transform, as well as several families of explicit solutions. A large part of this work is devoted to a rigorous approximation result that allows to project the longtime dynamics of the limit equation into that of the cubic NLS equation on a box of finite size.
Longtime instability and unbounded Sobolev orbits for some periodic nonlinear Schrödinger equations
, 2012
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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.
On the radius of analyticity of solutions to the cubic Szegő equation, in preparation
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Nondispersive solutions to the L2critical halfwave equation
 Arch. Ration. Mech. Anal
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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.
An improvement on the BrézisGallouët technique for 2D NLS and 1D halfwave equation
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