Absolute and convective instabilities of temporally oscillating flows

The theory of absolute and convective instabilities is extended to the spatially homogeneous, temporally oscillating case. A linear initial-boundary-value problem for small localised disturbances superimposed on an oscillatory basic state is treated by applying Fourier transform in space, Floquet decomposition in time and Laplace transform in time. The dispersion relation function of the problem is given in terms of the temporal Floquet exponents. The asymptotic evaluation of the solution, expressed as an inverse Fourier-Laplace integral, is obtained by applying the formalism developed in the stationary case. A collision criterion for the absolute instability and a causality condition for spatially amplifying waves are formulated in terms of the temporal Floquet exponents. We show that the oscillatory part of the asymptotics of wave packets and spatially amplifying waves is generally quasi-periodic in time. The theory is illustrated with two examples. In the first one, a scalar parabolic PDE is investigated completely on absolute instability. Second example treats exact oscillating solutions of the non-linear Schrodinger equation. We show that all such solutions are absolutely unstable.