Nonlinear Schrödinger equation and frequency saturation

Carles, Rémi

doi:10.2140/apde.2012.5.1157

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Nonlinear Schrödinger equation and frequency saturation

Rémi Carles

Vol. 5 (2012), No. 5, 1157–1173

DOI: 10.2140/apde.2012.5.1157

Abstract

We propose an approach that permits to avoid instability phenomena for the nonlinear Schrödinger equations. We show that by approximating the solution in a suitable way, relying on a frequency cut-off, global well-posedness is obtained in any Sobolev space with nonnegative regularity. The error between the exact solution and its approximation can be measured according to the regularity of the exact solution, with different accuracy according to the cases considered.

Keywords

nonlinear Schrödinger equation, well-posedness, approximation

Mathematical Subject Classification 2010

Primary: 35Q55

Secondary: 35A01, 35B30, 35B45, 35B65

Milestones

Received: 8 December 2011

Revised: 15 February 2012

Accepted: 20 March 2012

Published: 29 December 2012

Authors

Rémi Carles
CNRS & Université Montpellier 2 UMR 5149, Mathématiques, CC051 Place Eugène Bataillon 34095 Montpellier France

Vol. 5, No. 5, 2012