Length scale, quasiperiodicity, resonances, separatrix crossings, and chaos in the weakly relativistic Zakharov equations

Abstract

Nonlinear saturation of unstable solutions to the weakly relativistic, one-dimensional Zakharov equations is considered in this paper. In order to perform the analysis, two quantities are introduced. One of them, ${\mathrm{\ensuremath{\rho}}}_{\mathrm{*}}$, is proportional to the initial energy of the high-frequency field, and the other is the basic wave vector of the low-frequency perturbing mode k=2\ensuremath{\pi}/L, with L as the length scale. With these quantities it becomes possible to identify a number of regions on a ${\mathrm{\ensuremath{\rho}}}_{\mathrm{*}}$ versus k parametric plane. For very small values of ${\mathrm{\ensuremath{\rho}}}_{\mathrm{*}}$, steady-state solutions become unstable when k is also very small. In this case ion-acoustic dynamics is found to be unimportant and the system is numerically shown to be approximately integrable, even if k is below a critical value where the solutions are not simply periodic. For larger values of ${\mathrm{\ensuremath{\rho}}}_{\mathrm{*}}$ the unstable wave vectors also become larger and the ion-acoustic fluctuations turn into active dynamical modes of the system, driving a transition to chaos, which follows initial inverse pitchfork bifurcations. The transition includes resonant and quasiperiodic features; separatrix crossing phenomena are also found. The influence of relativistic terms on the chaotic dynamics is studied in the context of the Zakharov equations; it is shown that relativistic terms generally enhance the instabilities of the system, therefore anticipating the transition.