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

Abstract

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' $q\text{-nonextensive}$ statistics. In particular, the previous linear theory of Langmuir oscillations in EP plasmas [Saberian and Esfandyari-Kalejahi, Phys. Rev. E 87, 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 Schrödinger (NLS) equation with a nonlocal nonlinear term $\propto \mathcal{P}\int |\varphi ({\xi }^{\prime },\tau ){|}^{2}d{\xi }^{\prime }\varphi /(\xi -{\xi }^{\prime })$ [where $\mathcal{P}$ denotes the Cauchy principal value, $\varphi$ 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 $1/3<q\lesssim 3/5$ 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 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.