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… 

On certain Hamiltonian systems related to the cubic Szegő equation

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

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

Emphasising nonlinear behaviors for cubic coupled Schrödinger systems

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

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

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

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

Almost reducibility and oscillatory growth of Sobolev norms

. 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

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

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

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

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 41 REFERENCES

Explicit formula for the solution of the Szeg

We consider the cubic Szeg\"o equation i u_t=Pi(|u|^2u) in the Hardy space on the upper half-plane, where Pi is the Szeg\"o projector on positive frequencies. It is a model for totally non-dispersive

Invariant tori for the cubic Szegö equation

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

Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation

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.

Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation with a convolution potential

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

Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds

We prove Strichartz estimates with fractional loss of derivatives for the Schrödinger equation on any Riemannian compact manifold. As a consequence we infer low regularity local well-posedness

An Inverse Problem for Self-adjoint Positive Hankel Operators

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

The cubic nonlinear Schr\"odinger equation in two dimensions with radial data

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)

On differential structures of polynomial spaces in control theory

A valuable number of works has been published about Hurwitz and Schur polynomials in order to known better their properties. For example it is known that the sets of Hurwitz and Schur polynomials are

Integrals of Nonlinear Equations of Evolution and Solitary Waves

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

Operators, Functions, and Systems: An Easy Reading

Together with the companion volume by the same author, Operators, Functions, and Systems: An Easy Reading. Volume 2: Model Operators and Systems, Mathematical Surveys and Monographs, Vol. 93, AMS,