论文标题
带有三波相互作用的Schrödinger系统的站立波
Standing waves for a Schrödinger system with three waves interaction
论文作者
论文摘要
我们研究了具有三个波相互作用的非线性schrödinger方程系统的站立波，作为等离子体中拉曼扩增的模型。我们考虑了质量关键和质量危害的政权，我们证明了基态的存在以及同步的质量崩溃行为。此外，我们表明，在相关的考奇流动下，基集稳定。此外，在质量优势的环境中，我们构建了一个激发态，该态与强烈不稳定的驻波相对应。此外，精确绘制了激发态的半平整限制行为。最后，通过对激发态能量的精致控制，我们提供了足够的条件，以证明对相应的库奇问题的全球存在或炸毁解决方案。
We study standing waves for a system of nonlinear Schrödinger equations with three waves interaction arising as a model for the Raman amplification in a plasma. We consider the mass-critical and mass-supercritical regimes and we prove existence of ground states along with a synchronised mass collapse behavior. In addition, we show that the set of ground states is stable under the associated Cauchy flow. Furthermore, in the mass-supercritical setting we construct an excited state which corresponds to a strongly unstable standing wave. Moreover, a semi-trivial limiting behavior of the excited state is drawn accurately. Finally, by a refined control of the excited state's energy, we give sufficient conditions to prove global existence or blow-up of solutions to the corresponding Cauchy problem.
