Nonlinear evolution of an arbitrary density perturbation in a cold homogeneous unmagnetized plasma

An exact and general analytical solution describing the nonlinear evolution of a large amplitude plasma oscillation initiated by an arbitrary density perturbation which can be expressed as a Fourier series in x is found. This general solution is first used to reproduce the earlier results of R. C. Davidson and P. P. Schram [Nucl. Fusion 8, 183 (1968)] and J. M. Dawson [Phys. Rev. 113, 383 (1959)] using appropriate initial conditions and later applied to derive the space time evolution for a square wave and a triangular wave. It is found that the addition of a second harmonic increases the wave breaking limit of the fundamental mode. Analytical results pertaining to this two mode case have been verified using one-dimensional particle-in-cell simulation.