Nonlinear dynamics of driven relativistic electron plasma waves

We have examined the nonlinear dynamics associated with beat-wave ({Δ}{ω},{Δ}k) generation of long-wavelength plasma waves ({Δ}k{≤}ωₚ/c) in the presence of a strong ({δ}n/n-0.15 to 0.75) short-wavelength density ripple [kᵢ-(5 to 130) {Δ}k] using the relativistic Lagrangian-oscillator model. Two cases are considered: time-varying detuning ratio (ωₚ/{Δ}{ω}) and time-varying laser intensity I. In the absence of the plasma ripple, it is found that the Lagrangian-oscillator motion contains half-harmonic components in an Arnold tonguelike parameter space (ωₚ/{Δ}{ω},vosc/c) centered around ωₚ/{Δ}{ω}{}0.5. The effect of the ripple is twofold: (a) It lowers the minimum driver strength needed to access the half-harmonic parameter region around ωₚ/{Δ}{ω}{}0.5, and (b) it makes a second parameter region available, centered around ωₚ/{Δ}{ω}{}2.0. Although the Lagrangian model exhibits further period doubling followed by a transition to chaos when a time-varying laser intensity is used, wave breaking sets in after the first bifurcation, thereby limiting the validity of the model. The origin of the first period doubling, however, is found to be linked to the stability of an equivalent Mathieu equation to 1/2 subharmonic resonances. Finally, a particle-in-cell-code simulation shows spatial wave-number peaks displaced by {Δ}k/2 on both sides of the driver frequencies, giving support to the idea that the first bifurcation behavior may be observable in an experiment.