From the Zakharov system to the NLS equation on the torus

The NLS equation can be derived from the Zakharov system in a singular limit. The goal of this paper is to add some new aspects to the existing analysis about this approximation. We outline the differences between the situation $x\in\mathbb{R}$ and $x\in\mathbb{T}= \mathbb{R}/(2\pi\mathbb{Z})$. We explain the difficulties occurring in the construction of higher order approximations and use these approximations to improve the approximation rate. Moreover, we point out that the standard validity proof requires a smallness condition which was not stated in the existing literature.