  We outline a general theory for the analysis of flow-distributed standing and
travelling wave patterns in one-dimensional, open plug-flows of oscillatory
chemical media. We treat both the amplitude and phase dynamics of small and
large-amplitude waves, considering both travelling and stationary waves on an
equal footing and emphasizing features that are generic to a variety of kinetic
models. We begin with a linear stability analysis for constant and periodic
boundary forcing, drawing attention to the implications for systems far from a
Hopf bifurcation. Among other results, we show that for systems far from a Hopf
bifurcation, the first absolutely unstable mode may be a stationary wave mode.
We then introduce a non-linear formalism for studying both travelling and
stationary waves and show that the wave forms and their amplitudes depend on a
single reduced transport parameter. Our formalism sheds light on cases where
neither the linearized analyis nor the kinematic theory of phase waves give an
adequate description, and it can be applied to study some of the more complex
types of bifurcations (Canards, period-doublings, etc.) in open flow systems.

Pattern Formation and Solitons