ABSOLUTE AND CONVECTIVE INSTABILITIES, FRONT VELOCITIES AND GLOBAL MODES IN NONLINEAR-SYSTEMS

We study the existence of self-sustained saturated solutions of the re al Ginzburg-Landau equation subject to a boundary condition at x = 0; such solutions are called nonlinear global (NG) modes. The NG instabil ity referring to the existence of these solutions is rigorously determ ined and the scaling behavior of the NG modes close to threshold is de rived. The NG instability is first compared to the linear concept of c onvective/absolute (C/A) instability characterizing whether the impuls e 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 u nstable, or absolutely unstable. The growth size of the NG modes is sh own to be proportional to epsilon-(1/2) when NG and A instabilities ex ist simultaneously, epsilon being the criticality parameter, whereas a In(1/epsilon) scaling is found when the NG instability occurs while t he flow is C unstable or linearly stable. The nonlinear convective/abs olute (NC/NA) instability defined Chomaz (1992) by considering, in inf inite homogeneous domains, whether the front separating a bifurcated s tate 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 co incide. Similar results are presented for supercritical bifurcating sy stems, for the ''van der Pol-Duffing'' system, and for a transcritical model. In all the cases, the A instability is only a sufficient condi tion for the existence of an NG mode, and these simple models demonstr ate that a system may be nonlinearly absolutely unstable whereas it is linearly convectively unstable. This property should be generic if on e accepts the conjecture that the selected front velocity is always la rger than the linear front velocity. Response to a constant forcing ap plied at the origin is also studied. It is shown that in the NG region , the system possesses intrinsic dynamics which cannot be removed by t he 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.