Quasi-periodic solutions of nonlinear wave equations with quasi-periodic forcing

Abstract

This work focuses on one-dimensional (1D) quasi-periodically forced nonlinear wave equations. This means studying u t t − u x x + μ u + ε ϕ ( t ) h ( u ) = 0 , μ > 0 , with Dirichlet boundary conditions, where ε is a small positive parameter, ϕ ( t ) is a real analytic quasi-periodic function in t with frequency vector ω = ( ω 1 , ω 2 … , ω m ) and the nonlinearity h is a real analytic odd function of the form h ( u ) = η 1 u + η 2 r ̄ + 1 u 2 r ̄ + 1 + ∑ k ≥ r ̄ + 1 η 2 k + 1 u 2 k + 1 , η 1 , η 2 r ̄ + 1 ≠ 0 , r ̄ ∈ N . It is shown that, under a suitable hypothesis on ϕ ( t ) and h , there are many quasi-periodic solutions for the above equation via KAM theory.