On the Validity of the degenerate Ginzburg—Landau equation

Abstract

The Ginzburg–Landau equation which describes nonlinear modulation of the amplitude of the basic pattern does not give a good approximation when the Landau constant (which describes the influence of the nonlinearity) is small. In this paper a derivation of the so‐called degenerate (or generalized) Ginzburg–Landau (dGL)‐equation is given. It turns out that one can understand the dGL‐equation as an example of a normal form of a co‐dimension two bifurcation for parabolic PDEs. The main body of the paper is devoted to the proof of the validity of the dGL as an equation whose solution approximate the solution of the original problem. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

Rise and Fall of Periodic Patterns for a Generalized Klausmeier–Gray–Scott Model

“…For the general (complex) cubic Ginzburg-Landau equation, the Busse balloon may be non-existent, or its boundary may consist of certain Hopf bifurcations [54]. The structure of the Busse balloon becomes less simple, and for instance includes co-dimension 2 corners, for (extended) cubic-quintic Ginzburg-Landau equations that occur near the transition of super-to subcritical bifurcations [14,30,83,84]. In fact, the analysis in [30] indicates that there may even be bounded Busse balloons in this case.…”

Localized Turing patterns in nonlinear optical cavities

“…where b 3 and b 4 are given by (66), and (67), respectively, b (part) is the particular solution given by (68) and (71), and the K i are real constants. By evaluating the solution as X → −∞, we see that…”

“…In the appendix, we show how (5) can be deduced from the normal form in steady state. Finally, rigorous argument for the derivation of (5) are given in [68].…”

Instabilities and Dynamics of Weakly Subcritical Patterns

“…In the following we refer to this equation as the cubic-quintic Ginzburg-Landau equation or GL35 for short. This equation was first proposed in [2] and studied in [8,9]; its validity was analyzed rigorously in [3]. A generalization to two-dimensional systems with anisotropy, for example, for instabilities in a nematic liquid crystal, is given in [4].…”