SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 119-137, February 2024.
Abstract. The logarithmic Schrödinger equation (LogSE) has a logarithmic nonlinearity [math] that is not differentiable at [math]. Compared with its counterpart with a regular nonlinear term, it possesses richer and unusual dynamics, though the low regularity of the nonlinearity brings about significant challenges in both analysis and computation. Among very limited numerical studies, the semi-implicit regularized method via regularizing [math] as [math] to overcome the blowup of [math] at [math] has been investigated recently in the literature. With the understanding of [math] we analyze the nonregularized first-order implicit-explicit scheme for the LogSE. We introduce some new tools for the error analysis that include the characterization of the Hölder continuity of the logarithmic term, and a nonlinear Grönwall’s inequality. We provide ample numerical results to demonstrate the expected convergence. We position this work as the first to study the direct linearized scheme for the LogSE as far as we can tell.

中文翻译：

---

对数薛定谔方程一阶 IMEX 格式的误差分析

SIAM 数值分析杂志，第 62 卷，第 1 期，第 119-137 页，2024 年 2 月。
摘要。对数薛定谔方程 (LogSE) 具有在 [math] 处不可微分的对数非线性 [math]。与具有正则非线性项的对应项相比，它具有更丰富和不寻常的动力学特性，尽管非线性项的低正则性给分析和计算带来了巨大的挑战。在非常有限的数值研究中，最近文献中研究了通过将[math]正则化为[math]来克服[math]在[math]上的爆炸的半隐式正则化方法。通过对[数学]的理解，我们分析了 LogSE 的非正则化一阶隐式-显式方案。我们引入了一些用于误差分析的新工具，包括对数项的 Hölder 连续性的表征以及非线性 Grönwall 不等式。我们提供了充足的数值结果来证明预期的收敛。据我们所知，我们将这项工作定位为第一个研究 LogSE 直接线性化方案的工作。