一种求解含聚焦非线性项的阻尼非线性薛定谔方程的显式无条件稳定数值方法
An explicit unconditionally stable numerical method for solving damped nonlinear Schrödinger equations with a focusing nonlinearity
AI总结：
本文提出一种显式、无条件稳定的时间分裂正弦谱方法求解阻尼聚焦非线性薛定谔方程，数值实验表明三次或五次阻尼可阻止爆聚，线性阻尼需超过阈值，并可推广至三维BEC动力学模拟。
摘要：
本文介绍了时间分裂正弦谱（TSSP）方法的一种扩展，用于求解阻尼聚焦非线性薛定谔方程（NLS）。该方法具有显式、无条件稳定和时间横向不变性。此外，若在NLS中加入线性阻尼项，该方法能精确保持波函数归一化的衰减率。针对二维三次聚焦非线性薛定谔方程，我们进行了广泛的数值测试，分别考虑线性、三次或五次阻尼项。数值结果表明，五次或三次阻尼总能阻止爆聚，而线性阻尼仅当阻尼参数$\dt$大于阈值$\dt_{\rm th}$时才能阻止爆聚。我们注意到，该方法还可用于求解带五次阻尼项的三维Gross-Pitaevskii方程，以模拟玻色-爱因斯坦凝聚体（BEC）的坍缩与爆炸动力学。
英文摘要：
This paper introduces an extension of the time-splitting sine-spectral (TSSP) method for solving damped focusing nonlinear Schrödinger equations (NLS). The method is explicit, unconditionally stable and time transversal invariant. Moreover, it preserves the exact decay rate for the normalization of the wave function if linear damping terms are added to the NLS. Extensive numerical tests are presented for cubic focusing nonlinear Schrödinger equations in 2d with a linear, cubic or a quintic damping term. Our numerical results show that quintic or cubic damping always arrests blowup, while linear damping can arrest blowup only when the damping parameter $\dt$ is larger than a threshold value $\dt_{\rm th}$. We note that our method can also be applied to solve the 3d Gross-Pitaevskii equation with a quintic damping term to model the dynamics of a collapsing and exploding Bose-Einstein condensate (BEC).
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