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38
ON THE GLOBAL WELLPOSEDNESS OF ENERGYCRITICAL SCHRÖDINGER EQUATIONS IN CURVED SPACES
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Explicit formula for the solution of the Szegö equation on the real line and applications
, 2011
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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.
Resonant dynamics for the quintic nonlinear Schrödinger equation
 Ann. Inst. H. Poincaré Anal. Non Linéaire
, 2012
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On scattering for the quintic defocusing nonlinear Schrödinger equation on R×T2
 Comm. Pure and Appl. Math
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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.
Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus
 Communications in Partial Differential Equations
"... Abstract It is shown that plane wave solutions to the cubic nonlinear Schrödinger equation on a torus behave orbitally stable under generic perturbations of the initial data that are small in a highorder Sobolev norm, over long times that extend to arbitrary negative powers of the smallness parame ..."
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Cited by 6 (1 self)
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Abstract It is shown that plane wave solutions to the cubic nonlinear Schrödinger equation on a torus behave orbitally stable under generic perturbations of the initial data that are small in a highorder Sobolev norm, over long times that extend to arbitrary negative powers of the smallness parameter. The perturbation stays small in the same Sobolev norm over such long times. The proof uses a Hamiltonian reduction and transformation and, alternatively, Birkhoff normal forms or modulated Fourier expansions in time.