  A kinetic theory for quantum Langmuir waves interacting nonlinearly with
quantum ion-acoustic waves is derived. The formulation allows for a statistical
analysis of the quantum correction to the Zakharov system. The influence of a
background random phase on the modulational instability is given. In the
coherent case, the effect of the quantum correction is to reduce the growth
rate. Moreover, in the classical limit, a bifurcation develops in the
dispersion curves due to the presence of partial coherence. However, the
combined effect of partial coherence and a quantum correction may give rise to
an increased modulational instability growth rate, as compared to the classical
case. The results may be of significance in dense astrophysical plasmas and
laboratory laser-plasma systems.
