论文标题
在具有$χ^{（2）} $非线性的光学介质中产生的不均匀NLS系统，第I部分：Dynamics
A system of inhomogeneous NLS arising in optical media with a $χ^{(2)}$ nonlinearity, part I : Dynamics
论文作者
论文摘要
我们研究了一个不均匀的非线性schrödinger方程系统，该方程在具有$χ^{（2）} $非线性的光学介质中出现。这种非线性，其局部强度受到尖端的空间调制，$χ^{（2）} \ sim | x | x |^{ - α} $，其中$α> 0 $，可以通过空间不均匀的杆子诱导。我们的第一步是建立与系统相关的媒介gagliardo-Nirenberg类型的不平等。这使我们能够在初始数据上确定导致全局时间解决方案存在的必要条件。通过利用非线性无穷大的空间衰减，我们证明了全球解决方案的质量质临界状态中的非统治能量散射。这些解决方案的初始数据位于质量能阈值以下，无论系统是质量共振还是非质量共振剂。最后，我们提供了在质量和非质量共振案例中具有质量关键和质量质量非线性的非质量爆炸解决方案的标准。
We study a system of inhomogeneous nonlinear Schrödinger equations that emerge in optical media with a $χ^{(2)}$ nonlinearity. This nonlinearity, whose local strength is subject to a cusp-shaped spatial modulation, $χ^{(2)}\sim |x|^{-α}$ where $α> 0$, can be induced by spatially non-uniform poling. Our first step is to establish a vectorial Gagliardo--Nirenberg type inequality related to the system. This allows us to identify the necessary conditions on the initial data that lead to the existence of global in time solutions. By exploiting the spatial decay at infinity of the nonlinearity, we demonstrate the non-radial energy scattering in the mass-supercritical regime for global solutions. These solutions have initial data that lie below a mass-energy threshold, regardless of whether the system is mass-resonant or non-mass resonant. Lastly, we provide the criteria for the existence of non-radial blow-up solutions with mass-critical and mass-supercritical nonlinearities in both mass and non-mass resonance cases.
