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 o n 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 solut ion, expressed as an inverse Fourier-Laplace integral, is obtained by applying the formalism developed in the stationary case. A collision c riterion for the absolute instability and a causality condition for sp atially amplifying waves are formulated in terms of the temporal Floqu et exponents. We show that the oscillatory part of the asymptotics of wave packets and spatially amplifying waves is generally quasi-periodi c in time. The theory is illustrated with two examples. In the first o ne, a scalar parabolic PDE is investigated completely on absolute inst ability. Second example treats exact oscillating solutions of the non- linear Schrodinger equation. We show that all such solutions are absol utely unstable.