论文标题
在$ 1 $ d的立方NLS上
On the $1$d cubic NLS with a non-generic potential
论文作者
论文摘要
我们认为$ 1D $ CUCIC非线性schrödinger方程，其外部潜在$ V $是非类型的。在数据上没有任何奇偶校验假设，但假设相关的schrödinger运算符的零能量共振是奇怪的，甚至是奇数，我们证明了小型解决方案的全局定量界限和渐近学。 首先，我们对扭曲的傅立叶变换（DFT）的基础进行了简单的修改，以解决由于存在共振和不存在溶液对称性而在零能量下的（可能的）不连续性。然后，我们对（修改）非线性光谱分布的低频结构进行了精致分析，并在非生成电位的设置中采用平滑估计值。
We consider the $1d$ cubic nonlinear Schrödinger equation with an external potential $V$ that is non-generic. Without making any parity assumption on the data, but assuming that the zero energy resonance of the associated Schrödinger operator is either odd or even, we prove global-in-time quantitative bounds and asymptotics for small solutions. First, we use a simple modification of the basis for the distorted Fourier transform (dFT) to resolve the (possible) discontinuity at zero energy due to the presence of a resonance and the absence of symmetry of the solution. We then use a refined analysis of the low frequency structure of the (modified) nonlinear spectral distribution, and employ smoothing estimates in the setting of non-generic potentials.
