LENGTH SCALE, QUASI-PERIODICITY, RESONANCES, SEPARATRIX CROSSINGS, AND CHAOS IN THE WEAKLY RELATIVISTIC ZAKHAROV EQUATIONS

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, p*, 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 pi/L, with L as the length scale. With these quantities it becomes possible to identify a number of regions on a pr versus k parametric plane. For very small values of p*, 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 p* 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.