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

This work focuses on one-dimensional (1D) quasi-periodically forced nonlinear wave equations. This means studying utt−uxx+μu+εϕ(t)h(u)=0,μ>0, with Dirichlet boundary conditions, where εε is a small positive parameter, ϕ(t)ϕ(t) is a real analytic quasi-periodic function in tt with frequency vector ω=(ω1,ω2…,ωm)ω=(ω1,ω2…,ωm) and the nonlinearity hh is a real analytic odd function of the form h(u)=η1u+η2r̄+1u2r̄+1+∑k≥r̄+1η2k+1u2k+1,η1,η2r̄+1≠0,r̄∈N. It is shown that, under a suitable hypothesis on ϕ(t)ϕ(t) and hh, there are many quasi-periodic solutions for the above equation via KAM theory.