The nonlinear Schrödinger equation as a resonant normal form

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.