VOLUME 71, NUMBER 13 PH YSICAL REVIEW LETTERS 27 SEPTEMBER 1993 Comment on "Absolute and Convective Instabilities in Nonlinear Systems" When a spatially extended system goes unstable, the ensuing dynamics depends sensitively on whether the sys- tem is convectively unstable [in which case perturbations grow in time but are convected away fast enough that they die at each fixed position in the (lab) frame con- sidered] or absolutely unstable (in which case there exists a perturbation and a location where the perturbation does not decay). The distinction between the two cases for infinitesimal disturbances is well understood; such a linear stability analysis captures most of the essential physics near a supercritical (continuous) bifurcation. Re- cently, Chomaz [ I ] studied the nonlinear convective (NLC) versus absolute (NLA) instability near a subcriti- cal (discontinuous) bifurcation for a simple equation that derives from a free-energy-like (Lyapunov) function. The purpose of this Comment is to point out that the case studied by Chomaz is quite restrictive, since it relies on the existence of a unique front separating the basic state from the bifurcating state. In the general case there is a continuum of bifurcating states and an ensuing continu- um of fronts, so the problem of selection must be faced. The situation was discussed earlier by two of us [2] in a general investigation of front and pulse propagation near subcritical bifurcations. The extension to systems not governed by a Lyapunov function is particularly relevant for the study of nonlinear stability of open hydrodynamic flows or of systems with traveling waves. As a simple model for dynamics near a subcritical bi- furcation, Chomaz [1] studied the real equation 8, A+ Up&, A =c) A+ pA+A The nonlinear stability properties depend on the response to disturbances of finite extent and amplitude. For & p & 0 Eq. (1) admits two homogeneous stable states, Ap =0 and 22&0. To study the nonlinear stability of the Ap state it suffices to consider a front solution join- ing the state A2 for x — ~ with the state Ap for x ~, in the symmetrical (Up =0) frame where the UpB„A term is absent. If the front speed i of this solu- tion is negative, an isolated droplet of the A~ state in a background of the Ap state shrinks; hence the Ap state is stable. If t. is positive, A2 droplets grow and the Ap state is (nonlinearly) unstable. Since for Up=0, Eq. (1) is governed by a Lyapunov function [B,A = — 15K/&f, X = fdx j(8„A) /2 — pA /2 — A /4+A /6]], the sign of v depends on the relative magnitude of L(Ap) and X(Az), and v =0 for p =pM =— —, '6 where X(Ap) =X(Az). In the unstable domain p & pM the instability in the Up frame is convective (NLC) for v — Up &0, and absolute (NLA) for v — Up) 0. When a Hopf bifurcation to traveling waves occurs, the amplitude dynamics near a subcritical bifurcation can be modeled by an extension of (1), the complex Ginzburg- Landau equation, which in the symmetrical (Up =0) frame reads B,A = (I +ic ) t)„A+ pA+ (1+ic3)A ~A ~ + ( — 1+ic5)A ~A ~ (2) M. van Hecke and W. van Saarloos Instituut Lorentz, Nieuwsteeg 18, 2311 SB Leiden, The Netherlands P. C. Hohenberg AT%, T Bell Labs, Murray Hill, New Jersey 07974 Received 18 January 1993 PACS numbers: 47. 20. Ft, 47. 20. Ky [1] J. M. Chomaz, Phys. Rev. Lett. 69, 1931 (1991). [2] W. van Saarloos and P. C. Hohenberg, Phys. Rev. Lett. 64, 743 (1990); Physica (Amsterdam) 56D, 303 (1992). [3] See, e.g. , P. Kolodner, Phys. Rev. Lett. 66, 1165 (1991); F. Daviaud, J. Hegseth, and P. Berge, Phys. Rev. Lett. 69, 2511 (1992). [4] L. M. Hocking, K. Stewartson, and 3. T. Stuart, 3. Fluid Mech. 51, 705 (1972). Here 2 is the complex valued amplitude, and the c's are real parameters associated with the linear (ci) and non- linear (c3, cs) dispersion. Equation (2) cannot be derived from a Lyapunov function, and contrary to (1) has a con- tinuum of bifurcating states. The surprising finding of Ref. [2] is that the stability properties of the state Ap are largely determined by the existence or absence of an exact nonlinear front solution with speed v (p, ci, c3, cs) that increases for increasing p and is zero for p =p3(ci, c3, cs). It is found [2] that ei- ther (a) this front solution exists and has positive v for some range p & p3 with p3 & 0; (b) for all p & 0 the front speed is negative (i.e. , p3 & 0); or (c) for p & 0 no non- linear front solution exists. In case (a) the behavior for p & p3 is similar to that found in the real equation when p & pM.. The state Ap is unstable, and the instability is NLA for i t — Up~ 0 and NLC for i ~ — Up &0. For p & p3, on the other hand, typically stationary pulse solutions exist, over a range p2 & p & p3 so although t. & 0, the state A p remains un- stable. Since the pulse velocity is in general zero, the in- stability is NLC for any Up) 0. For p & p2 the state Ap is stable. In case (b) the pulse region extends up to p =0, and for p ) 0 the stability properties are similar to those of a supercritical bifurcation with a front velocity i cc Jp. For case (c) less is known, but chaotically spreading front solutions as well as pulses have been found [2]. In some experiments [3], the latter structures help stabilize a sys- tem by absorbing small perturbations that are convected into them. It is an open question which regime is relevant for planar Poiseuille I]ow, where cl — — 0.4 and c3= 6 [4] but e~ is not known. 2162 0031-9007/93/71(13)/2162(1) $06.00 1993 The American Physical Society