Uniform Adiabatic Limit of Benney type Systems

In this paper we show that solutions of the cubic nonlinear Schr"odinger equation are asymptotic limit of solutions to the Benney system. Due to the special characteristic of the one-dimensional transport equation same result is obtained for solutions of the one-dimensional Zakharov and 1d-Zakharov-Rubenchik systems. Convergence is reached in the topology $L^2(\mathbb{R})\times L^2(\mathbb{R})$ and with an approximation in the energy space $H^1(\mathbb{R})\times L^2(\mathbb{R})$. In the case of the Zakharov system this is achieved without the condition $\partial_t n(x,0) \in \dot H^{-1}($\mathbb{R}$)$ for the wave component, improving previous results.