Open AccessThe nonlinear Schrödinger equation as a resonant normal form
Author(s) -
Dario Bambusi,
A. Carati,
Antonio Ponno
Publication year - 2002
Publication title -
discrete and continuous dynamical systems. series b
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.864
H-Index - 53
eISSN - 1553-524X
pISSN - 1531-3492
DOI - 10.3934/dcdsb.2002.2.109
Subject(s) - monochromatic color , wave packet , nonlinear system , nonlinear schrödinger equation , amplitude , physics , schrödinger equation , mathematical analysis , dispersive partial differential equation , mathematical physics , classical mechanics , mathematics , quantum mechanics , optics
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 E, we show that the corresponding solution consists of two non interacting wave packets, each one being described by a nonlinear Schrodinger 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