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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.
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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SCATTERING FOR NONLINEAR SCHRÖDINGER EQUATION UNDER PARTIAL HARMONIC CONFINEMENT
, 2013
"... ABSTRACT. We consider the nonlinear Schrödinger equation under a partial quadratic confinement. We show that the global dispersion corresponding to the direction(s) with no potential is enough to prove global in time Strichartz estimates, from which we infer the existence of wave operators thanks to ..."
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Cited by 3 (0 self)
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ABSTRACT. We consider the nonlinear Schrödinger equation under a partial quadratic confinement. We show that the global dispersion corresponding to the direction(s) with no potential is enough to prove global in time Strichartz estimates, from which we infer the existence of wave operators thanks to suitable vectorfields. Conversely, given an initial Cauchy datum, the solution is global in time and asymptotically free, provided that confinement affects one spatial direction only. This stems from anisotropic Morawetz estimates, involving a marginal of the position density.
Global existence, scattering and blowup for the focusing NLS on the hyperbolic space
, 2014
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