Modulation of a quantum positron acoustic wave

Amplitude modulation of a positron acoustic wave is considered in a four-component electron–positron plasma in the quantum magnetohydrodynamic regime. The important ingredients of this study are the inclusion of the particle exchange-correlation potential, quantum diffraction effects via the Bohm potential, and dissipative effect due to viscosity in the momentum balance equation of the charged carriers. A modified nonlinear Schrödinger equation is derived for the evolution of the slowly varying amplitude of the quantum positron acoustic wave by employing the standard reductive perturbation technique. Detailed analysis of the linear and nonlinear dispersions of the quantum positron acoustic wave is presented. For a typical parameter range, relevant to some dense astrophysical objects, it is found that the quantum positron acoustic wave is modulationally unstable above a certain critical wavenumber. Effects of the exchange-correlation potential and the Bohm potential in the wave dynamics are also studied. It is found that the quantum effect due to the particle exchange-correlation potential is significant in comparison to the effect due to the Bohm potential for smaller values of the carrier wavenumber. However, for comparatively larger values of the carrier wavenumber, the Bohm potential effect overtakes the effect of the exchange-correlation potential. It is found that the critical wavenumber for the modulation instability depends on the ratio of the equilibrium hot electron number density and the cold positron number density and on the ratio of the equilibrium hot positron number density and the cold positron number density. A numerical result on the growth rate of the modulation instability is also presented.