  In this paper we introduce a system coupling a nonlinear Schr\"odinger
equation with a system of viscoelasticity, modeling the interaction between
short and long waves, acting for instance on media like plasmas or polymers. We
prove the existence and uniqueness of local (in time) strong solutions and the
existence of global weak solutions for the corresponding Cauchy problem. In
particular we extend previous results in [Nohel \emph{et. al.}, Commun. Part.
Diff. Eq., 13 (1988)] for the quasilinear system of viscoelasticity. We finish
with some numerical computations to illustrate our results.
