Oscillatory relativistic motion of a particle in a power-law or sinusoidal-shaped potential well

We report on an analytical work about the one-dimensional undamped relativistic motion of a particle located in a potential well whose space dependence is given by a power law or a sinusoidal law. This analysis is of interest in the present-day research field on interaction of ultra-high-intensity laser pulses with plasmas. First, we consider the oscillating motion of a particle trapped inside a potential-energy profile of the form K‖x${\mathrm{\Vert }}^{\mathit{n}}$ where K is a constant, x is the space coordinate, and n is a positive real number. The cases n=1 and n=2 are emphasized since the former is related to the motion of electrons that exit an homogeneous electrically neutral plasma towards vacuum and the latter is related to the usual electron plasma oscillation. Second, we study the potential-energy profile cos(x) in connection with the acceleration of an electron inside an electron plasma wave. Both trapped and untrapped particle motions are considered. For all the potential-energy shapes, analytic expressions of both the period of the oscillating motion and the particle trajectory are provided. The particle motion in the weakly relativistic case is discussed. The acceleration length of a particle trapped inside a sinusoidal drifting wave is finally calculated.