Revisiting the Cauchy problem for the Zakharov-Rubenchik/Benney-Roskes system

In this paper, we revisit the Cauchy problem for the Zakharov-Rubenchik/Benney-Roskes system. Our method is based on the dispersive estimates and the suitable Bourgain's spaces. We then, obtain the local well-posedness of the solution with the main component $ψ$ belongs to $H^1(\mathbb{R}^d)$ ($d=2, 3$) which is actually the energy space corresponding to this component. Our result also suggests a potential approach to the problem of finding exact existence time scale for the solution of Benney-Roskes model in the context of water waves.