Absolute Parametric Instabilities in Inhomogeneous Plasmas

For one-dimensional three-wave parametric instabilities with oppositely directed group velocities for the decay waves in unbounded plasmas, it is known that for uniform coupling amplitude and constant wave-vector mismatch ($\frac{d\Sigma k_i}{\mathrm{dx}}=\mathrm{const}$) there cannot be an absolute instability. It is shown that this is an exceptional case, and that in general an absolute instability can be obtained in an inhomogeneous plasma with a sufficiently strong pump. A simple criterion is given for the threshold of the absolute instability.