  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.
