Chapter 15 The Ginzburg-Landau equation in its role as a modulation equation

Abstract

The Ginzburg–Landau equation (GLe) ∂tA=div[(I+iS)∇A]+ρA−cˆ|A|2A,with A(t,x)∈ℂ,t⩾0,x∈ℝd, appears in many different contexts, e.g., nonlinear optics with dissipation or the theory of superconductivity. In addition, it plays an important role as modulation equation and it serves as a simple mathematical model for studying the transition from regular to turbulent behavior when, for δ > 0 fixed, the dispersion parameters (S,Imcˆ)=s(I,−δ)withs≫1 are considered. The purpose of this work is to review the latter two aspects of the GLe. In a variety of pattern-forming systems the nonlinear modulations of the basic periodic pattern can be described by the solutions of the GLe. We discuss this for three classical hydrodynamical situations: the Rayleigh-Bénard convection, Poiseuille and the Taylor-Couette problem. Indeed, the GLe should be seen as a normal form or the lowest order expansion of a bifurcation equation in the context of a weakly unstable system when continuous spectrum moves over the imaginary axis upon changing an external parameter. In particular, this theory gives a natural and simple approach to the theory of sideband instabilities. We explain the classical derivation of the GLe as a modulation equation of an original partial differential equation ∂tu=Lu+N(u) and give the abstract setting which allows for the application of the Ginzburg–Landau formalism which is based on the ansatz u(t,x)≈UA(t,x)=ɛA(ɛ2t,ɛ(x−cgrt))ei(ωt+〈k,x〉)Φ+c.c.,where ɛ2 is the distance of the external parameter from its critical value. We then study the mathematical properties of the GLe which are relevant for the justification of the formalism. This includes a variety of special solution classes, a global semigroup theory in function spaces containing L∞(ℝd) as well as the construction of the global attractor. Finally we review the results which prove that the solutions A of the GLe inserted into the above ansatz provide good approximations for solutions u of the original system. In particular, the direct implications to hydrodynamical problems are discussed.