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 or der to perform the analysis, two quantities are introduced. One of the m, p, is proportional to the initial energy of the high-frequency fie ld, and the other is the basic wave vector of the low-frequency pertur bing mode k = 2 pi/L, with L as the length scale. With these quantitie s 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 show n 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 fluc tuations turn into active dynamical modes of the system, driving a tra nsition to chaos, which follows initial inverse pitchfork bifurcations . The transition includes resonant and quasiperiodic features; separat rix crossing phenomena are also found. The influence of relativistic t erms 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.