具有排斥性调和势的对数薛定谔方程的数值研究
Numerical study of the logarithmic Schrodinger equation with repulsive harmonic potential
AI总结：
本文通过广义透镜变换和非均匀网格上的标准分裂方法，研究具有排斥性调和势的对数薛定谔方程，并给出 Lie-Trotter 分裂的误差估计及数值实验。
摘要：
我们考虑具有对数非线性和排斥性调和势的薛定谔方程。根据方程参数的不同，解可能具有色散性，也可能不具有色散性。当色散发生时，其随时间呈指数速率进行。为了控制这一现象，我们通过广义透镜变换改变未知函数。该方法中和了可能的边界效应，并且可用于无势薛定谔方程的情形。在对数非线性正则化之后，我们在新方程上通过非均匀网格采用标准分裂方法。我们还讨论了幂非线性情形，并针对两种非线性情形给出了一阶 Lie-Trotter 分裂方法的误差估计相关结果。最后，报告了大量数值实验以研究这些方程的动力学行为。
英文摘要：
We consider the Schrodinger equation with a logarithmic nonlinearity and a repulsive harmonic potential. Depending on the parameters of the equation, the solution may or may not be dispersive. When dispersion occurs, it does with an exponential rate in time. To control this, we change the unknown function through a generalized lens transform. This approach neutralizes the possible boundary effects, and could be used in the case of the Schrodinger equation without potential. We then employ standard splitting methods on the new equation via a nonuniform grid, after the logarithmic nonlinearity has been regularized. We also discuss the case of a power nonlinearity and give some results concerning the error estimates of the first-order Lie-Trotter splitting method for both cases of nonlinearities. Finally extensive numerical experiments are reported to investigate the dynamics of the equations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
