Second Order Hamiltonian Equations on T∞ and Almost-Periodic Solutions

AbstractMotivated by problems arising in nonlinear PDE′s with a Hamiltonian structure and in high dimensional dynamical systems, we study a suitable generalization to infinite dimensions of second order Hamiltonian equations of the type ẍ = ∂xV, [x ∈ TN, ∂x ≡ (∂x1, ..., ∂xN)]. 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 ẍi = ƒi(x), i ∈ Zd, x ∈ TZd (endowed with the compact topology); the Hamiltonian structure is reflected by ƒ being a "generalized gradient." Such a result is derived under (suitable) analyticity assumptions on ƒi but without requiring any "smallness conditions.