On the long time behavior for solutions of semi-linear harmonic oscillator with small Cauchy data on R^d.

Benoit Grebert (1) , Rafik Imekraz (1) , Eric Paturel (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

We consider the semi-linear harmonic oscillator $$i\psi_t=(-\Delta +x^{2} +M)\psi +\partial_2 g(\psi,\bar \psi), \quad x\in \R^d,\ t\in \R$$ where $M$ is a Hermite multiplier and $g$ a smooth function globally of order at least three.\\ We prove that such a Hamiltonian equation admits, in a neighborhood of the origin, a Birkhoff normal form at any order and that, under generic conditions on $M$ related to the non resonance of the linear part, this normal form is integrable when $d=1$ and gives rise to simple dynamics (in particular bounded) when $d\geq 2$.\\ As a consequence we prove the almost global existence for solutions of the above equation with small Cauchy data.

Domaines: Equations aux dérivées partielles [math.AP]; Systèmes dynamiques [math.DS]

Benoit Grebert, Rafik Imekraz, Eric Paturel. On the long time behavior for solutions of semi-linear harmonic oscillator with small Cauchy data on R^d.. 2008. ⟨hal-00309695v2⟩