Supersonic limit for the Zakharov–Rubenchik system

Abstract

We study the asymptotic behavior of the solutions of Zakharov–Rubenchik system when appropriate parameters go to zero. Namely, we state weak and strong convergence results of these solutions to solutions of Zakharov system. The proof of the weak limit is a classical argument in the theory of compactness, whose main ingredient is the Aubin–Lions Theorem and the Ascoli Theorem. Strong limits are conveniently treated by decomposing the nonlinearities and using the Strichartz estimates associated with the group of the Schrödinger equation and the wave group.