L. Brevdo et Tj. Bridges, ABSOLUTE AND CONVECTIVE INSTABILITIES OF TEMPORALLY OSCILLATING FLOWS, Zeitschrift fur angewandte Mathematik und Physik, 48(2), 1997, pp. 290-309
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.