Decay instabilities for inhomogeneous plasmas: WKB analysis and absolute instability

A general approximate criterion is developed (via WKB analysis) for the absolute instability of two parametrically pumped decay waves with one-dimensional inhomogeneity. The absolute instability threshold pump power proves to be higher than that required for amplitude amplification R by a factor πk′l2/27R, when k′ is the phase mismatch dΣki/dx at the location of maximum instability and l−2 is the effective curvature of the necessarily second-order expansion about that point. Previous results are included as special cases.