DIFFERENTIAL FLOW INSTABILITY IN THE GINZBURG-LANDAU AND SWIFT-HOHENBERG APPROXIMATIONS

The effects of a differential flow of the components of a reaction-dif fusion system which is close to its stability boundary are described w ithin the long wavelength approximation. In the vicinity of the Hopf b ifurcation the system's evolution is governed by a complex Ginzburg-La ndau equation modified by a purely imaginary convective term. If the s ystem is near the zero real eigenvalue bifurcation, the governing equa tion is a modified Swift-Hohenberg equation, In both cases the homogen eous, stable reference steady state may be destabilized by the differe ntial flow. In the Ginzburg-Landau equation, the destabilization occur s as long as the flow velocity exceeds some critical value upsilon(cr) , which tends to zero as the system approaches the Hopf bifurcation. I n the modified Swift-Hohenberg equation, the flow has either a destabi lizing or stabilizing effect, depending on the sign of one of the syst em parameters. Destabilization occurs when the flow velocity exceeds s ome threshold; however in this case, the threshold remains finite even at the bifurcation point. In both Ginzburg-Landau and Swift-Hohenberg equations the differential flow instability produces traveling plane waves. The stability analysis shows that once a periodic plane wave is established, its spatial period remains unchanged over a finite range of the flow velocity and changes in discrete steps - the phenomenon o f 'wavenumber locking'. Wavenumber locking' is verified in numerical e xperiments with the Ginzburg-Landau equation. Near Hopf bifurcation, t he Benjamin-Feir instability may occur. In this case irregular traveli ng waves are found, but a regular component of the wave pattern surviv es. Depending on a parameter, the differential flow either promotes or deters the Benjamin-Feir instability. As a result, the increasing flo w may switch the periodic wave pattern into a irregular state or, conv ersely, may stabilize the previously induced irregular pattern and pro duce periodic waves.