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

Kurzfassung/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 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.