Extensive Properties of the Complex Ginzburg-Landau Equation

Abstract

We study the set of solutions of the complex Ginzburg-Landau equation in $\real^d, d<3$. We consider the global attracting set (i.e., the forward map of the set of bounded initial data), and restrict it to a cube $Q_L$ of side $L$. We cover this set by a (minimal) number $N_{Q_L}(\epsilon)$ of balls of radius $\epsilon$ in $\Linfty(Q_L)$. We show that the Kolmogorov $\epsilon$-entropy per unit length, $H_\epsilon =\lim_{L\to\infty} L^{-d} \log N_{Q_L}(\epsilon)$ exists. In particular, we bound $H_\epsilon$ by $\OO(\log(1/\epsilon)$, which shows that the attracting set is smaller than the set of bounded analytic functions in a strip. We finally give a positive lower bound: $H_\epsilon>\OO(\log(1/\epsilon))$