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 Schr\"odinger 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.
|Titolo:||The nonlinear Schrödinger equation as a resonant normal form|
|Autori interni:||BAMBUSI, DARIO PAOLO (Primo)|
CARATI, ANDREA (Secondo)
|Parole Chiave:||Nonlinear Schr¨odinger equation; general dispersive equations.|
|Settore Scientifico Disciplinare:||Settore MAT/07 - Fisica Matematica|
|Data di pubblicazione:||2002|
|Appare nelle tipologie:||01 - Articolo su periodico|