The existence of countably many stable cycles for a generalized cubic Schrödinger equation in a planar domain

We consider the boundary-value problem
$$ u_t+i\Delta u=\varepsilon(u-d|u|^2u), \qquad u\big|_{\partial \Omega}=0, $$
in the domain $\Omega=\{(x,y)\colon 0\leqslant x\leqslant 1,0\leqslant y\leqslant 1\}$, where $u$ is a complex-valued function, $\Delta$ is the Laplace operators, $0<\varepsilon\ll1$ and $d=1+ic_0$, $c_0\in\mathbb R$. We establish that it has countably many stable solutions that are periodic in $t$. We study the question of whether this phenomenon is preserved under a change of domain or boundary conditions.