LOW-DIMENSIONAL PHASE-LOCKED STATES IN THE ZAKHAROV EQUATIONS

In this paper we identify phase-locked states among the solutions of t he Zakharov equations. Locked states appear as resonant island chains in the appropriate Poincare plots, with the relevant surface of sectio n obtained by projecting out the full dynamical set on a subspace defi ned in terms of a pair of center-manifold variables. This pair allows an accurate canonical description of the system immediately after an i nverse pitchfork bifurcation destabilizes an initial homogeneous stead y state. If one is very close to the bifurcation point, nonlinear satu ration of the initial instability is provided by quasistatic integrabl e ion-acoustic fluctuations, but as one proceeds away from that point, resonant nonintegrable ion-acoustic fluctuations become gradually mor e important; we show that the phase-locked states result from those re sonant fluctuations. If one is not too far from the pitchfork bifurcat ion, locking is the stable asymptotic state of the interaction. As one moves farther away, locking exists only over long but finite amounts of time. In addition, the resonance separatrix appears to bring the fi rst chaotic activity into the system.