Plasma-wave generation in the beat-wave accelerator

We analytically study the generation of longitudinal plasma waves in an un- derdense plasma by two electromagnetic waves with frequency difference approx- imately equal to the plasma frequency, as envisioned in the plasma beat-wave accelerator concept of Tajima and Dawson. The relativistic electron fluid equa- tions describing driven electron oscillations with phase velocities near the speed of light in a cold, collisionless plasma are reduced to a single, approximate ordinary differential equation of a parametrically excited nonlinear oscillator. We give amplitude-phase equations describing the asymptotic solutions to this equation valid for plasma wave amplitudes below wave-breaking. We numerically compare the behavior of the asymptotic equations with that of the original equation and with particle simulation results. -.