Self-modulation of nonlinear Alfvén waves in a strongly magnetized relativistic electron-positron plasma

We study the self-modulation of a circularly polarized Alfvén wave in a strongly magnetized relativistic electron-positron plasma with finite temperature. This nonlinear wave corresponds to an exact solution of the equations, with a dispersion relation that has two branches. For a large magnetic field, the Alfvén branch has two different zones, which we call the normal dispersion zone (where dω/dk>0) and the anomalous dispersion zone (where dω/dk<0). A nonlinear Schrödinger equation is derived in the normal dispersion zone of the Alfvén wave, where the wave envelope can evolve as a periodic wave train or as a solitary wave, depending on the initial condition. The maximum growth rate of the modulational instability decreases as the temperature is increased. We also study the Alfvén wave propagation in the anomalous dispersion zone, where a nonlinear wave equation is obtained. However, in this zone the wave envelope can evolve only as a periodic wave train.