Extensive Properties of the Complex Ginzburg-Landau Equation

Abstract

We study the set of solutions of the complex Ginzburg-Landau equation in ${\Re }^d,d<3$. 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 $\OO(\log(1/ε)$, 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: $H_ε>\OO(\log(1/ε))$

24 pages

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