LUIGI CHIERCHIA , PAOLO PERFETTI
Second Order Hamiltonian Equations on ${\bf T}^\infty$ and Almost-Periodic
Solutions
(87K, Plain TeX)
ABSTRACT. Motivated by problems arising in nonlinear PDE's with a Hamiltonian
structure and in high dimensional systems, we study a suitable generalization
to infinite dimensions of second order Hamiltonian equations of the type
$\ddot x=\partial_xV,$ [$x\in {\bf T}^N,$ $\partial_x\equiv(\partial_{x_1},
\dots,\partial_{x_N})$]. Extending methods from quantitative perturbation
theory (Kolmogorov-Arnold-Moser theory, Nash-Moser implicit function theorem,
etc.) we construct uncountably many almost-periodic solutions for the infinite
dimensional system $\ddot x_i=f_i(x), i\in{\bf Z}^d, x\in{\bf T}^{\bf Z}^d$
(endowed with the compact topology); the Hamiltonian structure is reflected
by $f$ being a "generalized gradient". Such result is derived under (suitable)
analyticity assumptions on $f_i$ but without requiring any "smallness
conditions".