The Zakharov limit of Klein–Gordon–Zakharov like systems in case of analytic solutions

We give a new proof for the validity of the Zakharov approximation of the Klein–Gordon–Zakharov system under the assumption that the solutions of the Zakharov system are analytic in a strip in the complex plane. The approach is conceptually more simple than for solutions which are only finitely many times differentiable and so it can simply be applied to more general systems, too.