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ASYMPTOTIC BEHAVIOR OF THE NONLINEAR SCHRÖDINGER EQUATION WITH HARMONIC TRAPPING
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Energy cascades for NLS on the torus
 Discrete and Continuous Dynamical Systems. Series A
, 2012
"... ABSTRACT. We consider the nonlinear Schrödinger equation with cubic (focusing or defocusing) nonlinearity on the multidimensional torus. For special small initial data containing only five modes, we exhibit a countable set of time layers in which arbitrarily large modes are created. The proof relie ..."
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ABSTRACT. We consider the nonlinear Schrödinger equation with cubic (focusing or defocusing) nonlinearity on the multidimensional torus. For special small initial data containing only five modes, we exhibit a countable set of time layers in which arbitrarily large modes are created. The proof relies on a reduction to multiphase weakly nonlinear geometric optics, and on the study of a particular twodimensional discrete dynamical system. 1.
An explicit formula for the cubic Szegő equation
, 2013
"... We derive an explicit formula for the general solution of the cubic Szegő equation and of the evolution equation of the corresponding hierarchy. As an application, we prove that all the solutions corresponding to finite rank Hankel operators are quasiperiodic. ..."
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We derive an explicit formula for the general solution of the cubic Szegő equation and of the evolution equation of the corresponding hierarchy. As an application, we prove that all the solutions corresponding to finite rank Hankel operators are quasiperiodic.
Geometric Optics and Instability for NLS AND DAVEYSTEWARTSON MODELS
"... We study the interaction of (slowly modulated) high frequency waves for multidimensional nonlinear Schrödinger equations with gauge invariant powerlaw nonlinearities and nonlocal perturbations. The model includes the Davey–Stewartson system in its ellipticelliptic and hyperbolicelliptic variant ..."
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We study the interaction of (slowly modulated) high frequency waves for multidimensional nonlinear Schrödinger equations with gauge invariant powerlaw nonlinearities and nonlocal perturbations. The model includes the Davey–Stewartson system in its ellipticelliptic and hyperbolicelliptic variant. Our analysis reveals a new localization phenomenon for nonlocal perturbations in the high frequency regime and allows us to infer strong instability results on the Cauchy problem in negative order Sobolev spaces, where we prove norm inflation with infinite loss of regularity by a constructive approach.
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.
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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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.
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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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
The energy graph of the non–linear Schrödinger equation
 REND. LINCEI MAT. APPL
, 2013
"... We discuss the stability of a class of normal forms of the completely resonant non–linear Schrödinger equation on a torus described in [12]. The discussion is essentially combinatorial and algebraic in nature. ..."
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We discuss the stability of a class of normal forms of the completely resonant non–linear Schrödinger equation on a torus described in [12]. The discussion is essentially combinatorial and algebraic in nature.