  The nonlinear theory of Landau damping of electrostatic wave envelopes (WEs)
is revisited in a quantum electron-positron (EP) pair plasma. Starting from a
Wigner-Moyal equation coupled to the Poisson equation and applying the multiple
scale technique, we derive a nonlinear Schr{\"o}dinger (NLS) equation which
governs the evolution of electrostatic WEs. It is shown that the coefficients
of the NLS equation, including the nonlocal nonlinear term, which appears due
to the resonant particles having group velocity of the WEs, are significantly
modified by the particle dispersion. The effects of the quantum parameter $H$
(the ratio of the plasmon energy to the thermal energy densities), associated
with the particle dispersion, are examined on the Landau damping rate of
carrier waves, as well as on the modulational instability of WEs. It is found
that the Landau damping rate and the decay rate of the solitary wave amplitude
are greatly reduced compared to their classical values $(H=0)$.
