THE NONLINEAR KLEIN-GORDON EQUATION ON AN INTERVAL AS A PERTURBED SINE-GORDON EQUATION

We treat the nonlinear Klein-Gordon (NKG) equation as the Sine-Gordon (SG) equation, perturbed by a higher order term. It is proved that mos t small-amplitude finite-gap solutions of the SG equation, which satis fy either Dirichlet or Neumann boundary conditions, persist in the NKG equation and jointly form partial central manifolds, which are ''Lips chitz manifolds with holes''. Our proof is based on an analysis of the finite-gap solutions of the boundary problems for SG equation by mean s of the Schottky uniformization approach, and an application of an in finite-dimensional KAM-theory.