Parametric instabilities of a circularly polarized Alfven wave in a nondispersive relativistic electron-positron plasma

Dispersion relations of parametric instabilities of circularly polarized Alfven waves in a relativistic electron-positron plasma is investigated. In addition to the usual decay and beat instabilities, two new types of instabilities involving the parallel/anti-parallel propagating Alfven-like waves and the parallel propagating Langmuir-like wave are found to be destabilized when the plasma parameters satisfy the condition, $2omega_{\rm p}^2 > (omega_0+Omega_0)^2$, where $omega_{\rm p}^2$, $omega_0$, and $Omega_0$ respectively represent the plasma frequency, parent wave frequency, and the gyro-frequency. The maximum growth rates for these instabilities are derived analytically. Further, weakly relativistic effects are discussed for all the instabilities