Parametric decays of a left-handed circularly polarized Alfvén wave propagating along a constant background magnetic field in a relativistic thermal electron-positron plasma are studied by means of a one dimensional relativistic particle-in-cell simulation. Relativistic effects are included in the Lorentz equation for the momentum of the particles and in their thermal motion, by considering a Maxwell-Jüttner velocity distribution function for the initial condition. In the linear stage of the simulation, we find many instabilities that match the predictions of relativistic fluid theory. In general, the growth rates of the instabilities increase as the pump wave amplitude is increased, and decrease with a raise in the plasma temperatures. We have confirmed that for very high temperatures the Alfvén branch is suppressed, consistent with analytical calculations.
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López et al. (2014) studied this question.