Solutions of the classical and nonlocal Gross-Pitaevskii (GP) equation with a parabolic potential and a gain term are derived by using a second order nonisospectral Ablowitz-Kaup-Newell-Segur system and reduction technique of double Wronskians. Solutions of the classical GP equation show typical space-time localized characteristics. An interesting dynamics, solitons carrying an oscillating wave, are found with mathematical analysis and illustrations. Solutions of some nonlocal cases are also illustrated.

中文翻译：

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具有抛物线势的经典和非局部 Gross-Pitaevskii 方程的新动力学

具有抛物线势和增益项的经典和非局部 Gross-Pitaevskii (GP) 方程的解是通过使用二阶非等谱 Ablowitz-Kaup-Newell-Segur 系统和双 Wronskians 约简技术推导出来的。经典 GP 方程的解表现出典型的时空定域特征。通过数学分析和插图，发现了一种有趣的动力学，即携带振荡波的孤子。还说明了一些非本地情况的解决方案。