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

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• 作者：Rovinsky ; Arkady ; Malevanets ; Anatoly ; Menzinger ; Michael
• 关键词：PACS ; 47.10+g ; 47.70-n ; 47.20.Ky ; 47.20Tg
• 刊名：Physica D
• 出版年：1996
• 出版时间：September 1, 1996
• 年：1996
• 卷：95
• 期：3-4
• 页码：306-318
• 全文大小：982 K

文摘

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 center border=0 SRC=/images/glyphs/CDU.GIF>_cr, 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 switch the periodic wave pattern into a irregular state or, conversely, may stabilize the previously induced irregular pattern and produce periodic waves.