This notes studies the inhomogeneous non-linear Schrödinger equations with a harmonic potential

$$\begin{aligned} i\partial _tu +\Delta u-|x|^2u+|x|^{b}|u|^{p-1}u=0. \end{aligned}$$

Indeed, following the methods of Fukuizumi and Ohta (Differ Integral Equ 16(6):691–706, 2003), Ohta (Funccialaj Ekvacioj 61:135–143, 2018), the non-linear and strong instability of standing waves are obtained in the two different regimes \(b>0\) and \(b<0\).

中文翻译：

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一类具有谐波势的不均匀Schrödinger方程的驻波不稳定性

本笔记研究具有谐波电位的非齐次非线性Schrödinger方程

$$ \ begin {aligned} i \ partial _tu + \ Delta u- | x | ^ 2u + | x | ^ {b} | u | ^ {p-1} u = 0。\ end {aligned} $$

确实，按照Fukuizumi和Ohta（Differ Integral Equ 16（6）：691-706，2003），Ohta（Funccialaj Ekvacioj 61：135-143，2018）的方法，获得了非线性和强驻波不稳定性在两个不同的状态\（b> 0 \）和\（b <0 \）中。