A direct construction of a slow manifold for a semilinear wave equation of Klein–Gordon type

Abstract

We study a semilinear wave equation whose linear part corresponds to the linear Klein–Gordon equation in the non-relativistic limit, augmented with a nonlinearity that is Fréchet-differentiable over the complex numbers. We show that this equation possesses an almost invariant manifold in phase space that generalizes the slow manifold which is known to exist for finite-dimensional Galerkin truncations of the system. This manifold is shown to be almost invariant to any algebraic order and can be constructed in the Hs−1×Hs phase space of the equation uniformly in the order of the approximation. In particular, we prove that the dynamics on this “slow manifold” shadows orbits of the full system over a finite interval of time.