Abstract
A set of differential equations is set up to describe the temporal and spatial evolution of transverse and longitudinal waves under mode-mode coupling in a relativistic plasma embedded in an ambient magnetic field. The plasma consists of immobile protons, thermal electrons and cosmic ray protons. The cosmic rays are assumed to be streaming along the ambient magnetic field direction with a bulk velocity larger than the Alfven velocity. An Alfven wave couples with another transverse and one longitudinal wave. The temporal mode-mode coupling effects on the nonlinear behaviour of the Alfven wave (which is unstable using linear theory in the absence of mode-mode coupling) have been examined. Under certain conditions, it is found that the linear instability of the Alfven wave is 'quenched'. Upon inserting numerical parameters appropriate to the cosmic ray gas in the galactic disk it is shown that when the initial energy of perturbation is 10-2 of the ambient field energy, the rate of quenching is the same order as the linear e-folding time (10 approximately 102 yr when the streaming velocity of the cosmic ray is approximately 108 cm/sec).