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15
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.
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.
ASYMPTOTIC BEHAVIOR OF THE NONLINEAR SCHRÖDINGER EQUATION WITH HARMONIC TRAPPING
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1 LARGE TIME BLOW UP FOR A PERTURBATION OF THE CUBIC SZEGŐ EQUATION
"... Abstract. We consider the following Hamiltonian equation on a special manifold of rational functions, i∂tu=Π(u  2 u)+α(u1),α∈R, whereΠdenotes the Szegő projector on the Hardy space of the circleS 1. The equation withα=0 was first introduced by Gérard and Grellier in [6] as a toy model for totally ..."
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Cited by 2 (0 self)
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Abstract. We consider the following Hamiltonian equation on a special manifold of rational functions, i∂tu=Π(u  2 u)+α(u1),α∈R, whereΠdenotes the Szegő projector on the Hardy space of the circleS 1. The equation withα=0 was first introduced by Gérard and Grellier in [6] as a toy model for totally non dispersive evolution equations. We establish the following properties for this equation. Forα<0, any compact subset of initial data leads to a relatively compact subset of trajectories. Forα>0, there exist trajectories on which high Sobolev norms exponentially grow with time. hal00846626, version 1 19 Jul 2013 1.
BEHAVIOR OF A MODEL DYNAMICAL SYSTEM WITH APPLICATIONS TO WEAK TURBULENCE
"... Abstract. We experimentally explore solutions to a model Hamiltonian dynamical system recently derived to study frequency cascades in the cubic defocusing nonlinear Schrödinger equation on the torus. Our results include a statistical analysis of the evolution of data with localized amplitudes and ra ..."
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Cited by 1 (1 self)
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Abstract. We experimentally explore solutions to a model Hamiltonian dynamical system recently derived to study frequency cascades in the cubic defocusing nonlinear Schrödinger equation on the torus. Our results include a statistical analysis of the evolution of data with localized amplitudes and random phases, which supports the conjecture that energy cascades are a generic phenomenon. We also identify stationary solutions, periodic solutions in an associated problem and find experimental evidence of hyperbolic behavior. Many of our results rely upon reframing the
UNBOUNDED SOBOLEV TRAJECTORIES AND MODIFIED SCATTERING THEORY FOR A WAVE GUIDE NONLINEAR Schrödinger equation
, 2015
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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.