The Complex Ginzburg-landau Equation∗

Essential to the derivation of the Ginzburg-Landau equation is assumption that the spatial variables of the vector field U(x, y, t) are defined on a cylindrical domain. This means that (x, y) ∈ R ×Ω, where Ω ⊂ R is a open and bounded domain (and m ≥ 1, n ≥ 0), so that U : R ×Ω×R+ → R . The N ×N constant coefficient matrix Sμ is assumed to be non-negative, in the sense that all its eigenvalues are real and ≥ 0. Note that this implies that Sμ is not necessarily invertible. In most situations, Sμ will be trivial, i.e. the identity matrix. However, for instance when (1.1) describes a model in fluid dynamics, Sμ can have a zero eigenvalue. This eigenvalue is then associated to the equation that governs the conservation of mass, which does not have an explicit time derivative in it. The differential operators Lμ and Nμ are of course essential to the dynamics of (1.1). The linear operator Lμ is assumed to be elliptic; Nμ is a nonlinear operator of order less than LR. Neither Lμ nor Nμ is allowed to depend explicitly on x or t. However, both Lμ = Lμ(y) and Nμ = Nμ(y), i.e both operators may depend directly on the bounded variable y ∈ Ω. At this point we do not need to impose any additional constraints on Lμ or Nμ.