A Nekhoroshev-type theorem for the nonlinear Schrödinger equation on the torus

where $V$ is a typical smooth Fourier multiplier and $g$ is analytic in both variables. More precisely, we prove that if the initial datum is analytic in a strip of width $ρ > 0$ whose norm on this strip is equal to $ε$ , then if $ε$ is small enough, the solution of the nonlinear Schrödinger equation above remains analytic in a strip of width $ρ ∕ 2$ , with norm bounded on this strip by $C ε$ over a very long time interval of order $ε^{- σ | ln ε |^{β}}$ , where $0 < β < 1$ is arbitrary and $C > 0$ and $σ > 0$ are positive constants depending on $β$ and $ρ$ .

Keywords

Nekhoroshev theorem, nonlinear Schrödinger equation, normal forms

Mathematical Subject Classification 2010

Primary: 35B40, 35Q55, 37K55

Milestones

Received: 16 February 2011

Revised: 23 January 2013

Accepted: 28 February 2013

Published: 18 November 2013

Authors

Erwan Faou
INRIA and ENS Cachan Bretagne Avenue Robert Schumann F-35170 Bruz France

Benoît Grébert
Laboratoire de Mathématiques Jean Leray Université de Nantes 2 Rue de la Houssiniere F-44322 Nantes Cedex 3 France

Vol. 6, No. 6, 2013

Erwan Faou and Benoît Grébert

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors