Regularity, approximation and asymptotic dynamics for a generalized Ginzburg-Landau equation

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Abstract
In this paper, we study regularity and asymptotic dynamics of a generalized complex Ginzburg-Landau (GL) amplitude equation. We show that the solutions belong to a Gevrey class of regularity and are real analytic in the spatial variable. We use this to derive an adaptive method based on Galerkin approximation and show that it converges exponentially fast. We also show that the equation has a finite dimensional compact global attractor, and has at most two determining nodes. This result, which depends on regularity, implies that asymptotic behaviour can be determined from a small number of observations in physical space.