The existence of countably many stable cycles for a~generalized cubic Schr\"odinger equation in a~planar domain

A. Yu. Kolesov ; N. Kh. Rozov

Izvestiya. Mathematics , Tome 67 (2003) no. 6, pp. 1213-1242

Résumé

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.