  The nonlinear theory of amplitude modulation of electrostatic wave envelopes
in a collisionless electron-positron (EP) pair plasma is studied by using a set
of Vlasov-Poisson equations in the context of Tsallis' $q$-nonextensive
statistics. In particular, the previous linear theory of Langmuir oscillations
in EP plasmas [Phys. Rev. E {\bf87}, 053112 (2013)] is rectified and modified.
Applying the multiple scale technique (MST), it is shown that the evolution of
electrostatic wave envelopes is governed by a nonlinear Schr{\"o}dinger (NLS)
equation with a nonlocal nonlinear term $\propto
{\cal{P}}\int|\phi(\xi',\tau)|^2d\xi'\phi/(\xi-\xi') $ [where ${\cal P}$
denotes the Cauchy principal value, $\phi$ is the small-amplitude electrostatic
(complex) potential, and $\xi$ and $\tau$ are the stretched coordinates in MST]
which appears due to the wave-particle resonance. It is found that a subregion
$1/3<q\lesssim3/5$ of superextensivity $(q<1)$ exists where the carrier wave
frequency can turn over with the group velocity going to zero and then to
negative values. The effects of the nonlocal nonlinear term and the
nonextensive parameter $q$ are examined on the modulational instability (MI) of
wave envelopes as well as on the solitary wave solution of the NLS equation. It
is found that the modulated wave packet is always unstable (nonlinear Landau
damping) due to the nonlocal nonlinearity in the NLS equation. Furthermore, the
effect of the nonlinear Landau damping is to slow down the amplitude of the
wave envelope, and the corresponding decay rate can be faster the larger is the
number of superthermal particles in pair plasmas.
