Convergent and divergent perturbation series and the post-Minkowskian approximation scheme

The question of convergence and divergence of perturbation series for hyperbolic equations is considered, together with the closely related question of existence of analytic one-parameter families of solutions. It is shown that for semilinear equations there is a plentiful supply of such families so that perturbation series converge under reasonable conditions. It is then shown that for more general quasilinear equations this no longer holds. In the particular case of the Einstein equations it is found that the kind of perturbation series used in the post-Minkowskian approximation scheme does not in general converge. In the course of the proof of this last fact, it is shown that the constraint equations have the property of analytic local linearisation stability.