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

摘要

This work focuses on one-dimensional (1D) quasi-periodically forced nonlinear wave equations. This means studying u(u) - u(xx) + mu u + epsilon phi(t)h(u) = 0, mu > 0, with Dirichlet boundary conditions, where epsilon is a small positive parameter, phi(t) is a real analytic quasi-periodic function in t with frequency vector omega = (omega(1), omega(2) ..., omega(m)) and the nonlinearity h is a real analytic odd function of the form h(u) = eta(1)u + eta(2 (r) over bar +1)u(2 (r) over bar +1) + Sigma(k >=(r) over bar1) eta(2k+1)u(2k+1), eta(1), eta(2 (r) over bar +1) not equal 0, (r) over bar is an element of N. It is shown that, under a suitable hypothesis on phi(t) and h, there are many quasi-periodic solutions for the above equation via KAM theory.