A Nekhoroshev theorem for some infinite--dimensional systems

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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

Abstract
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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.

Keywords: Nekhoroshev theorem, infinite--dimensional systems., stability result.

Mathematics Subject Classification: Primary: 70K65, 70K70, 70K43, 70K40; Secondary: 34C29, 34C2.

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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