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The nonlinear theory of amplitude modulation of electrostatic wave envelopesin a collisionless electron-positron (EP) pair plasma is studied by using a setof Vlasov-Poisson equations in the context of Tsallis' $q$-nonextensivestatistics. In particular, the previous linear theory of Langmuir oscillationsin EP plasmas [Phys. Rev. E {\bf87}, 053112 (2013)] is rectified and modified.Applying the multiple scale technique (MST), it is shown that the evolution ofelectrostatic 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 wavefrequency can turn over with the group velocity going to zero and then tonegative values. The effects of the nonlocal nonlinear term and thenonextensive parameter $q$ are examined on the modulational instability (MI) ofwave envelopes as well as on the solitary wave solution of the NLS equation. Itis found that the modulated wave packet is always unstable (nonlinear Landaudamping) due to the nonlocal nonlinearity in the NLS equation. Furthermore, theeffect of the nonlinear Landau damping is to slow down the amplitude of thewave envelope, and the corresponding decay rate can be faster the larger is thenumber of superthermal particles in pair plasmas.

Nonlinear Landau damping and modulation of electrostatic waves in a nonextensive electron-positron-pair plasma