  We study standing waves for a system of nonlinear Schr\"odinger equations
with three waves interaction arising as a model for the Raman amplification in
a plasma. We consider the mass-critical and mass-supercritical regimes and we
prove existence of ground states along with a synchronised mass collapse
behavior. In addition, we show that the set of ground states is stable under
the associated Cauchy flow. Furthermore, in the mass-supercritical setting we
construct an excited state which corresponds to a strongly unstable standing
wave. Moreover, a semi-trivial limiting behavior of the excited state is drawn
accurately. Finally, by a refined control of the excited state's energy, we
give sufficient conditions to prove global existence or blow-up of solutions to
the corresponding Cauchy problem.
