On the global and singular dynamics of the 2D cubic nonlinear Schrödinger equation on cylinders

We consider the Cauchy problem associated to the focusing cubic nonlinear Schrödinger equation posed on a two dimensional cylindrical domain $R\times T_{\ell }$. We prove that localized transverse perturbations of an especial one-parameter family of bound states solutions $\left\{u_{\omega ,\ell }\right\}$, $\omega >-\frac{{\pi }^2}{{\ell }^2}$ can be extended globally in time. On the other hand, we establish the existence of solution in the energy space $H^1(R\times T_{\ell })$, with non-critical mass, that blows-up in finite time under the hypothesis of no growth in time of the directional $L_x^2$-norm of the solution when the periodic variable $y$ is localized. We also prove that a family of bound states $\left\{u_{\omega ,\ell }\right\}$ is not uniformly continuous from $H^s(R\times T_{\ell })$ into the space of continuous functions $C\left([0,T];H^s(R\times T_{\ell })\right)$, whenever $-1/2\le s<0$, including the regularity $s=-\frac{1}{2}$ for the non-uniformly continuous flow, unlike to the case of focusing cubic nonlinear Schrödinger equation on $R$.