科研之友
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摘要
We study the existence of self-sustained saturated solutions of the real Ginzburg-Landau equation subject to a boundary condition at x = 0; such solutions are called nonlinear global (NG) modes. The NG instability referring to the existence of these solutions is rigorously determined and the scaling behavior of the NG modes close to threshold is derived. The NG instability is first compared to the linear concept of convective/absolute (C/A) instability characterizing whether the impulse response of an unstable flow in an infinite domain is asymptotically damped or amplified at a fixed location. NG modes are shown to exist while at the same time the flow may be linearly stable, convectively unstable, or absolutely unstable. The growth size of the NG modes is shown to be proportional to epsilon-(1/2) when NG and A instabilities exist simultaneously, epsilon being the criticality parameter, whereas a In(1/epsilon) scaling is found when the NG instability occurs while the flow is C unstable or linearly stable.
The nonlinear convective/absolute (NC/NA) instability defined Chomaz (1992) by considering, in infinite homogeneous domains, whether the front separating a bifurcated state from the basic state moves downstream or upstream, is determined using van Saarloos and Hohenberg (1992) results for the selected front velocity. Remarkably, the NA domain and the NG domain are shown to coincide. Similar results are presented for supercritical bifurcating systems, for the ''van der Pol-Duffing'' system, and for a transcritical model. In all the cases, the A instability is only a sufficient condition for the existence of an NG mode, and these simple models demonstrate that a system may be nonlinearly absolutely unstable whereas it is linearly convectively unstable. This property should be generic if one accepts the conjecture that the selected front velocity is always larger than the linear front velocity.
Response to a constant forcing applied at the origin is also studied. It is shown that in the NG region, the system possesses intrinsic dynamics which cannot be removed by the forcing, By contrast, the behavior of a nonlinear spatial amplifier is observed in a domain larger than the NC region. NC instability is only a sufficient condition to trigger the system with forcing.
- 出版日期1997-10-1
