In this paper, one-dimensional (1D) nonlinear wave equationswith periodic boundary conditions are considered; V is a periodic smooth or analytic function and the nonlinearity f is an analytic function vanishing together with its derivative at u = 0. It is proved that for "most" potentials V (x), the above equation admits small-amplitude periodic or quasi-periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system. The proof is based on an infinite dimensional KAM theorem which allows for multiple normal frequencies. * The research was partly supported by NNSF of China and by the italian CNR grant # 211.01.31. 1 s ( F z 0 Da,ρ(r,s),O + n∈Z + F zn Da,ρ(r,s),O nāe nρ ). Notational Remark In what follows, only the indices r, s and the set O will change while a,ā, ρ will be kept fixed, therefore we shall usually denote X F ā,ρ Da,ρ(r,s),O by X F r,s,O ,D a,ρ (r, s) by D(r, s) and F Da,ρ(r,s),O by F r,s,O . 3 Dot stands for the time derivatives d/dt. 4 The norm · Da,ρ(r,s),O for scalar functions is defined in (2.3). For vector (or matrix-valued) functions G : Da,ρ(r, s) × O → C m , (m < ∞) is similarly defined as G Da,ρ(r,s),O = m i=1 Gi Da,ρ(r,s),O (for the matrix-valued case the sum will run over all entries). 14 Recall the definition of Pi in (4.6). 15 Recall the definition of N in (4.1).
