The nonlinear Schrödinger equation as a resonant normal form

Dario Bambusi; A. Carati; A. Ponno

doi:10.3934/dcdsb.2002.2.109

Article Contents

2002, Volume 2, Issue 1: 109-128. Doi: 10.3934/dcdsb.2002.2.109

This issue Previous Article Finite element analysis and approximations of phase-lock equations of superconductivity Next Article Identification of modulated rotating waves in pattern-forming systems with O(2) symmetry

The nonlinear Schrödinger equation as a resonant normal form

1.
Dipartimento di Matematica “F. Enriques”, Universita di Milano, Via Saldini 50, 20133 Milano
2.
Dipartimento di Matematica, Università degli studi di Milano, Via Saldini 50, 20133 Milano
3.
Dipartimento di Fisica "Galileo Galilei", Università di Padova, Via Marzolo 8, 35131 Padova

Received: March 31, 2001

Revised: June 30, 2001

Published: January 2002

Abstract Related Papers Cited by

Abstract

Averaging theory is used to study the dynamics of dispersive equations taking the nonlinear Klein Gordon equation on the line as a model problem: For approximatively monochromatic initial data of amplitude $\epsilon$, we show that the corresponding solution consists of two non interacting wave packets, each one being described by a nonlinear Schrödinger equation. Such solutions are also proved to be stable over times of order $1/ \epsilon^2$. We think that this approach puts into a new light the problem of obtaining modulations equations for general dispersive equations. The proof of our results requires a new use of normal forms as a tool for constructing approximate solutions.

Keywords:

Nonlinear Schrödinger equation,
general dispersive equations.

Mathematics Subject Classification: 37K55, 37K40.

Citation:

Article Metrics

HTML views() PDF downloads(75) Cited by(0)

The nonlinear Schrödinger equation as a resonant normal form

Abstract

Access History

Article Metrics

Access History

Other Articles By Authors

Catalog

The nonlinear Schrödinger equation as a resonant normal form

Abstract

Access History

SHARE

Article Metrics

Access History

Other Articles By Authors

Catalog

Export File

Citation

Format

Content