Characteristic features of the dynamics of the Ginzburg–Landau equation in a plane domain

We study the boundary value problem $w_t=\varkappa_0\Delta w+\varkappa_1w-\varkappa_2w|w|^2$, $w|_{\partial\Omega_0}=0$ in the domain $\Omega_0=\bigl\{(x,y)\:0\leq x\leq l_1,0\leq y\leq l_2\bigr\}$. Here, $w$ is a complex-valued function, $\Delta$ is the Laplace operator, and $\varkappa_j$, $j=0,1,2$, are complex constants with $\mathrm{Re}\varkappa_j>0$. We show that under a rather general choice of the parameters $l_1$ and $l_2$, the number of stable invariant tori in the problem, as well as their dimensions, grows infinitely as $\mathrm{Re}\varkappa_0\to0$ and $\mathrm{Re}\varkappa_1\to0$.