February  2002, 2(1): 109-128. doi: 10.3934/dcdsb.2002.2.109

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, Italy

Received  April 2001 Revised  July 2001 Published  November 2001

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.
Citation: Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete and Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109
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