# On the growth of Sobolev norms for the cubic Szegő equation

@inproceedings{Grard2014OnTG,
title={On the growth of Sobolev norms for the cubic Szegő equation},
author={Patrick G{\'e}rard and Sandrine Grellier},
year={2014}
}
• Published 2014
• Mathematics
The large time behavior of solutions to Hamiltonian partial differential equations is an important problem in mathematical physics. In the case of finite dimensional Hamiltonian systems, many features of the large time behavior of trajectories are described using the topology of the phase space. For a given infinite dimensional systems, several natural phase spaces, with different topologies, can be chosen, and the large time properties may strongly depend on the choice of such topologies. For…
The main purpose of this Ph.D. thesis is to study the long time behavior of solutionsto some Hamiltonian PDEs, i∂_t u=X_H (u), including global existence, growth of high Sobolev norms, scattering and
This paper is devoted to the study of large time bounds for the Sobolev norms of the solutions of the following fractional cubic Schrodinger equation on the torus : $$i \partial_t u = |D|^\alpha The aim of this work is to propose a study of various nonlinear behaviors for a system of two coupled cubic Schrodinger equations. Depending on the choice of the spatial domain, we highlight • Mathematics • 2022 . For 1D quantum harmonic oscillator perturbed by a time quasi-periodic quadratic form of ( x, − i ∂ x ), we show its almost reducibility. The growth of Sobolev norms of solution is described based We consider the following Hamiltonian equation on the L^2 Hardy space on the circle S^1 ,$$i\partial\_ t u = \Pi(|u|^ 2 u) + \alpha(u|1) , \alpha \in\mathbb{R} ,$$where \Pi is the Szeg\H{o} • Mathematics • 2016 The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. This was complemented by the question as to how We consider the following wave guide nonlinear Schrödinger equation, WS$$\begin{aligned} (i\partial _t+\partial _{xx}-\vert D_y\vert )U=\vert U\vert ^2U\ \end{aligned}$$(i∂t+∂xx-|Dy|)U=|U|2Uon the • Mathematics • 2017 This monograph is an expanded version of the preprint arXiv:1402.1716 or hal-00943396v1. It is devoted to the dynamics on Sobolev spaces of the cubic Szego equation on the circle {\mathbb S} ^1, • Mathematics • 2016 Using suitable modified energies we study higher order Sobolev norms’ growth in time for the nonlinear Schrödinger equation (NLS) on a generic 2d or 3d compact manifold. In 2d we extend earlier ## References SHOWING 1-10 OF 32 REFERENCES We consider the cubic defocusing nonlinear Schrödinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. We consider the following Hamiltonian equation on a special manifold of rational functions, $i\p\_tu=\Pi(|u|^2u)+\al (u|1),\ \al\in\R,$ where \Pi  denotes the Szeg\H{o} projector on the Hardy • Mathematics • 2010 AbstractWe continue the study of the following Hamiltonian equation on the Hardy space of the circle,$$i\partial_tu=\Pi(|u|^2u), where Π denotes the Szegö projector. This equation can be seen as
• Physics, Mathematics
• 2014
The theme of this monograph is the nonlinear Schrodinger equation. This equation models slowly varying wave envelopes in dispersive media and arises in various physical systems such as water waves,
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• 2010
We consider the cubic defocusing nonlinear Schrödinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes.
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Fix $s>1$. Colliander, Keel, Staffilani, Tao and Takaoka proved in \cite{CollianderKSTT10} the existence of solutions of the cubic defocusing nonlinear Schr\"odinger equation in the two torus with
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Forum of Mathematics, Pi
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We consider the cubic nonlinear Schrödinger equation posed on the spatial domain $\mathbb{R}\times \mathbb{T}^{d}$. We prove modified scattering and construct modified wave operators for small
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For a sequence {αn} 1=0 , we consider the Hankel operator �, realised as the infinite matrix in l 2 with the entries αn+m. We consider the subclass of such Hankel operators defined by the "double
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• 2007
We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schr\"odinger equation $iu_t + \Delta u = \pm |u|^2 u$ for large spherically symmetric L^2_x(\R^2)
In Section 1 we present a general principle for associating nonlinear equations evolutions with linear operators so that the eigenvalues of the linear operator integrals of the nonlinear equation. A