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 , with non-critical mass, that blows-up in finite time under the hypothesis of no growth in time of the directional -norm of the solution when the periodic variable is localized. We also prove that a family of bound states is not uniformly continuous from into the space of continuous functions , whenever , including the regularity for the , unlike to the case of focusing cubic nonlinear Schrödinger equation on .

中文翻译：

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圆柱上二维三次非线性薛定谔方程的全局和奇异动力学

我们考虑与二维圆柱域上提出的聚焦三次非线性薛定谔方程相关的柯西问题。我们证明了一种特殊的单参数束缚态解族的局部横向扰动可以及时扩展到全局。另一方面，我们建立了能量空间中具有非临界质量的解的存在性，当周期变量为本地化。我们还证明，无论何时，包括 的正则性，一系列束缚态从连续函数的空间中都不是一致连续的，这与将三次非线性薛定谔方程集中在 上的情况不同。