Stability of nonlinear parametric-decay interactions in finite homogeneous plasma

Abstract

We prove the existence and uniqueness of a stable steady state of stimulated backscattering from a bounded homogeneous lossless interaction region, with boundary conditions corresponding to a steady incident pump wave-mode Poynting flux and zero-flux input to the decay wave modes. In steady state, once the excitation of the interaction region exceeds a certain critical value, the boundary value problem is characterized by a finite number of eigenvalues, and associated nontrivial eigenfunctions equilibria of the system, corresponding to mutually distinct states of anomalous reflection of the pump wave. A stability analysis of these equilibria with respect to small phase and amplitude perturbations reveals that (i) in the vicinity of the nonfundamental equilibria the phase perturbations exhibit singularities, preventing phase locking from occurring, and (ii) in the vicinity of the fundamental equilibrium both the phase and amplitude perturbations asymptotically vanish. A WKBJ phase-integral stability condition is derived to show that growing normal modes of the amplitude-perturbation boundary-value problem cannot propagate in the potential formed by the field of the depleted, spatially inhomogeneous, pump of the fundamental equilibrium.