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

Abstract

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