# 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} }

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…

## 22 Citations

### On certain Hamiltonian systems related to the cubic Szegő equation

- Mathematics
- 2015

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…

### On the growth of high Sobolev norms for certain one-dimensional Hamiltonian PDEs

- Mathematics
- 2015

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 :
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### On the growth of Sobolev norms of solutions of the fractional defocusing NLS equation on the circle

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### Emphasising nonlinear behaviors for cubic coupled Schrödinger systems

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### Almost reducibility and oscillatory growth of Sobolev norms

- 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…

### The Cubic Szeg\h{o} Equation with a Linear Perturbation

- Mathematics
- 2015

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}…

### Geometric Numerical Integration 871 Workshop : Geometric Numerical Integration Table of Contents

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- 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…

### Unbounded Sobolev trajectories and modified scattering theory for a wave guide nonlinear Schrödinger equation

- Mathematics
- 2015

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…

### The Cubic Szego Equation and Hankel Operators

- 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 ${\mathbb S} ^1$, …

### A P ] 2 9 Ju l 2 01 6 ON THE GROWTH OF SOBOLEV NORMS FOR NLS ON 2 d AND 3 d MANIFOLDS

- 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…

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