Quasi-periodic solutions for class of Hamiltonian partial differential equations with fixed constant potential

Abstract

We consider Hamiltonian partial differential equations u_tt +|∂_x|u+ σu = f(u), x ∈ T, t ∈ℝ, with periodic boundary conditions, where f(u) is a real-analytic function of the form f(u) = u⁵ + o(u⁵) near u = 0, σ ∈ (0, 1) is a fixed constant, and \mathbb{T}=ℝ/\mathrm{2}\pi \mathbb{Z}T= R/2πZ. A family of quasi-periodic solutions with 2-dimensional are constructed for the equation above with σ ∈ (0, 1)\ \mathbb{Q}. The proof is based on infinite-dimensional KAM theory and partial Birkhoff normal form.

Key words： Dense frequency; quasi-periodic solution; Birkhoff normal form

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