D. Bambusi, A. Ponno
Resonance, Metastability and Blow up in FPU
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ABSTRACT. We consider the FPU model with nonlinearity starting with terms of
order $n\geq 3$. We compute the resonant normal form in the region
where only one low frequency modes is excited and deduce rigorous
results on the correspondence between the dynamics of the normal form
and that of the complete system. As $n$ varies, we give a criterion in
order to deduce whether the FPU phenomenon (formation of a metastable
packet of modes) is present or not. The criterion is that, if the
normal form equation has smooth solutions then the FPU phenomenon is
present, while it is absent if the solutions of the normal form
equations have blow up in a finite time. In particular the
phenomenon should be present for $n\leq 6$ and absent for $n\geq 7$.