On the validity of the Ginzburg-Landau equation

Abstract

Summary, The famous Ginzburg-Landau equation describes nonlinear amplitude modulations of a wave perturbation of a basic pattern when a control parameter R lies in the unstable region O(e 2) away from the critical value Rc for which the system loses stability. Here e > 0 is a small parameter. G-Us equation is found for a general class of nonlinear evolution problems including several classical problems from hydrodynamics and other fields of physics and chemistry. Up to now, the rigorous derivation of G-Us equation for general situations is not yet completed. This was only demonstrated for special types of solutions (steady, time periodic) or for special problems (the Swift-Hohenberg equation). Here a mathematically rigorous proof of the validity of G-Us equation is given for a general situation of one space variable and a quadratic nonlinearity. Validity is meant in the following sense. For each given initial condition in a suitable Banach space there exists a unique bounded solution of the initial value problem for G-Us equation on a finite interval of the O(1/eZ)-long time scale intrinsic to the modulation. For such a finite time interval of the intrinsic modulation time scale on which the initial value problem for G-Us equation has a bounded solution, the initial value problem for the original evolution equation with corresponding initial conditions, has a unique solution O(e 2) -close to the approximation induced by the solution of G-L's equation. This property guarantees that, for rather general initial conditions on the intrinsic modulation time scale, the behavior of solutions of G-Us equation is really inherited from solutions of the original problem, and the other way around: to a solution of G-Us equation corresponds a nearby exact solution with a relatively small error.

Propagation of hexagonal patterns near onset

“…The properties of Ginzburg-Landau equations that are needed to make this statement more precise are attractivity [17,40] and validity of the approximation [9,23,39,44]. In this appendix, we state these results in the context of the Ginzburg-Landau system (3.13).…”

“…To derive an explicit expression for the right-hand side of (3.11), we follow the strategy in Eckmann & Wayne [16] and Haragus & Schneider [21], and relate the reduced equation on the center manifold to the Ginzburg-Landau equation [23,39] of the underlying PDE (1.1).…”

Amplitude Equations for Locally Cubic Nonautonomous Nonlinearities

“…The modes in our context are given by the Fourier series expansion with respect to the eigenfunctions of the corresponding linearized operator. For deterministic systems the theory is rigorously understood even for spatially extended systems (see, e.g., [KSM92,vH91] for the first results). However, there is a lack of results for stochastic systems.…”

Singularly Perturbed and Nonlocal Modulation Equations for Systems with Interacting Instability Mechanisms

“…Introducing transversal y-dimensions will merely increase the technical difficulties. The validity of the Ginzburg-Landau equation (1.2) for systems like (2.1) has been proved in [12]. As in the introduction, we define the neutral curve {Reµ(k, R) = 0}.…”

“…The above derivation of the nonlocal system (1.6) is not completely rigorous. In order to improve this, one should work with the Fourier transformψ of ψ, the solution of (2.1), and interpret it as a distribution; see for instance [12] and [3].…”