Growth rates of envelope modulations of electromagnetic waves in relativistic temperature electron-positron plasmas, stimulated by weak or finite phonon damping

We study the modulational instability of an electron-positron plasma with relativistic temperatures and phonon damping. It is shown that when the phonon damping is O (1) or O (ε), a modulational instability appears even for classical temperatures. When the phonon damping is O (ε2)), we find that in the non-relativistic limit, previous results are recovered, and for ultrarelativistic temperatures, the instability occurs in a wider band around the relativistic plasma frequency