Vol. 6, No. 6, 2013

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

Erwan Faou and Benoît Grébert

Vol. 6 (2013), No. 6, 1243–1262

We prove a Nekhoroshev type theorem for the nonlinear Schrödinger equation

iut = Δu + V u + ūg(u,ū),x Td,

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 ρ2, with norm bounded on this strip by Cε over a very long time interval of order εσ| ln ε|β , where 0 < β < 1 is arbitrary and C > 0 and σ > 0 are positive constants depending on β and ρ.

Nekhoroshev theorem, nonlinear Schrödinger equation, normal forms
Mathematical Subject Classification 2010
Primary: 35B40, 35Q55, 37K55
Received: 16 February 2011
Revised: 23 January 2013
Accepted: 28 February 2013
Published: 18 November 2013
Erwan Faou
INRIA and ENS Cachan Bretagne
Avenue Robert Schumann
F-35170 Bruz
Benoît Grébert
Laboratoire de Mathématiques Jean Leray
Université de Nantes
2 Rue de la Houssiniere
F-44322 Nantes Cedex 3