Dynamics of nonlinearly excited plasma waves

In an underdense plasma a large-amplitude plasma oscillation may be produced by the beating of two electromagnetic waves with a frequency difference approximately equal to the plasma frequency. In the spatially one-dimensional, cold, and collisionless plasma the large-amplitude plasma oscillation is limited by the nonlinearity caused by relativistic effects. In this paper a simple nonlinear equation, resembling the original equation of Rosenbluth and Liu [Phys. Rev. Lett. 29, 701 (1972)], is derived in the weak beat power limit from the fully relativistic fluid model proposed by Sprangle, Sudan, and Tang [Appl. Phys. Lett. 45, 375 (1984); Phys. Fluids 28, 1974 (1985)]. This equation also contains the effects of the relativistic transverse motion. Its analytical solution, describing the plasma oscillation dynamics, is given in a closed form by using Jacobian elliptic functions. The analytical computations are compared with numerical computations. Finally the fully relativistic equation, describing free plasma oscillations, is studied analytically.