REGULARITY, APPROXIMATION AND ASYMPTOTIC DYNAMICS FOR A GENERALIZED GINZBURG-LANDAU EQUATION

In this paper, we study regularity and asymptotic dynamics of a genera lized complex Ginzburg-Landau (GL) amplitude equation. We show that th e solutions belong to a Gevrey class of regularity and are real analyt ic in the spatial variable. We use this to derive an adaptive method b ased on Galerkin approximation and show that it converges exponentiall y fast. We also show that the equation has a finite dimensional compac t global attractor, and has at most two determining nodes. This result , which depends on regularity, implies that asymptotic behaviour can b e determined from a small number of observations in physical space.