In this paper, we study the nonlinear Schrödinger equation with focusing point nonlinearity in dimension one. First, we establish a scattering criterion for the equation based on Kenig–Merle’s compactness-rigidity argument. Then we prove the energy scattering below and above the mass-energy threshold. We also describe the dynamics of solutions with data at the ground state threshold. Finally, we prove a blow-up criteria for the equation with initial data with arbitrarily large energy.

中文翻译：

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具有高于质能阈值的点非线性的一维 NLS 的爆炸和散射

在本文中，我们研究了具有一维聚焦点非线性的非线性薛定谔方程。首先，我们基于 Kenig-Merle 的紧凑性-刚性论证为方程建立散射准则。然后我们证明低于和高于质能阈值的能量散射。我们还用基态阈值的数据描述了解决方案的动态。最后，我们证明了具有任意大能量的初始数据的方程的爆炸标准。