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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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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.
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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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.
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.
We consider the cubic nonlinear Schrödinger equation
, 2013
"... Plane wave solutions to the cubic nonlinear Schrödinger equation on a torus have recently been shown to behave orbitally stable. Under generic perturbations of the initial data that are small in a highorder Sobolev norm, plane waves are stable over long times that extend to arbitrary negative pow ..."
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Plane wave solutions to the cubic nonlinear Schrödinger equation on a torus have recently been shown to behave orbitally stable. Under generic perturbations of the initial data that are small in a highorder Sobolev norm, plane waves are stable over long times that extend to arbitrary negative powers of the smallness parameter. The present paper studies the question as to whether numerical discretizations by the splitstep Fourier method inherit such a generic longtime stability property. This can indeed be shown under a condition of linear stability and a nonresonance condition. They can both be verified if the time stepsize is restricted by a CFL condition in the case of a constant plane wave. The proof first uses a Hamiltonian reduction and transformation and then modulated Fourier expansions in time. It provides detailed insight into the
Dynamics in Geometric Dispersive Equations and the Effects of Trapping, Scattering and Weak Turbulence
, 2014
"... One of the major success stories in analysis over the past couple of decades is the deep and detailed insight into the qualitative properties of solutions to nonlinear dispersive PDE from Mathematical Physics which has been gained through application of techniques from harmonic analysis, spectral th ..."
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One of the major success stories in analysis over the past couple of decades is the deep and detailed insight into the qualitative properties of solutions to nonlinear dispersive PDE from Mathematical Physics which has been gained through application of techniques from harmonic analysis, spectral theory, the calculus of variations, and dynamical systems. In this way, our understanding of the nonlinear waves which characterize the dynamics of various systems in quantum physics, general relativity, optics, and fluid mechanics (just to name a few) has increased enormously over a remarkably short period. This understanding extends to questions of local and global wellposedness, lowregularity solutions, singularity formation, asymptotic behaviour, the existence and stability of special solutions (such as solitons, or various threshold solutions), and the role such special solutions play in the general dynamics. Progress has been such that it could be said, very roughly speaking, that our mathematical comprehension of nonlinear dispersive PDE which are (a) posed on Euclidean space, and (b) of a subcritical (roughly, conserved quantities provide some natural control over the size of solutions), or even (due to groundbreaking advances of the last few years) critical nature, is now rather good. On the other hand, it is just as reasonable to say that nonlinear wave equations outside of this category are still quite poorly understood, due to inherent new difficulties. For equations posed on compact domains