Differential flow instability in the Ginzburg-Landau and Swift-Hohenberg approximations

Abstract

The effects of a differential flow of the components of a reaction-diffusion system which is close to its stability boundary are described within the long wavelength approximation. In the vicinity of the Hopf bifurcation the system's evolution is governed by a complex Ginzburg-Landau equation modified by a purely imaginary convective term. If the system is near the zero real eigenvalue bifurcation, the governing equation is a modified Swift-Hohenberg equation. In both cases the homogeneous, stable reference steady state may be destabilized by the differential flow. In the Ginzburg-Landau equation, the destabilization occurs as long as the flow velocity exceeds some critical value vcr, which tends to zero as the system approaches the Hopf bifurcation. In the modified Swift-Hohenberg equation, the flow has either a destabilizing or stabilizing effect, depending on the sign of one of the system parameters. Destabilization occurs when the flow velocity exceeds some 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 of ‘wavenumber locking’. ‘Wavenumber locking’ is verified in numerical experiments with the Ginzburg-Landau equation. Near Hopf bifurcation, the Benjamin-Feir instability may occur. In this case irregular traveling waves are found, but a regular component of the wave pattern survives. Depending on a parameter, the differential flow either promotes or deters the Benjamin-Feir instability. As a result, the increasing flow may swithc the periodic wave pattern into a irregular state or, conversely, may stabilize the previously induced irregular pattern and produce periodic waves.