March  2006, 5(1): 125-146. doi: 10.3934/cpaa.2006.5.125

A Nekhoroshev theorem for some infinite--dimensional systems

1. 

Dipartimento di Matematica, Università di Tor Vergata, via della ricerca scientifica, 00133, Roma, Italy

Received  May 2005 Revised  November 2005 Published  December 2005

We study the persistence for long times of the solutions of some infinite--dimensional discrete hamiltonian systems with formal hamiltonian $\sum_{i=1}^\infty h(A_i) + V(\varphi),$ $(A,\varphi)\in \mathbb R^{\mathbb N}\times \mathbb T^{\mathbb N}.$ $V(\varphi)$ is not needed small and the problem is perturbative being the kinetic energy unbounded. All the initial data $(A_i(0), \varphi_i(0)),$ $i\in \mathbb N$ in the phase--space $\mathbb R^{\mathbb N} \times \mathbb T^{\mathbb N},$ give rise to solutions with $|A_i(t) - A_i(0)|$ close to zero for exponentially--long times provided that $A_i(0)$ is large enough for $|i|$ large. We need $\frac{\partial h}{\partial A_i}(A_i(0))$ unbounded for $i\to+\infty$ making $\varphi_i$ a fast variable the greater is $i,$ the faster is the angle $\varphi_i$ (avoiding the resonances). The estimates are obtained in the spirit of the averaging theory reminding the analytic part of Nekhoroshev--theorem.
Citation: Paolo Perfetti. A Nekhoroshev theorem for some infinite--dimensional systems. Communications on Pure & Applied Analysis, 2006, 5 (1) : 125-146. doi: 10.3934/cpaa.2006.5.125
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