The logarithmic nonlinearity has been used in many partial differential equations (PDEs) for modeling problems in various applications. Due to the singularity of the logarithmic function, it introduces tremendous difficulties in establishing mathematical theories, as well as in designing and analyzing numerical methods for PDEs with such nonlinearity. Here, we take the logarithmic Schrödinger equation (LogSE) as a prototype model. Instead of regularizing f(ρ) =ln ρ in the LogSE directly and globally as being done in the literature, we propose a local energy regularization (LER) for the LogSE by first regularizing F(ρ) = ρln ρ − ρ locally near ρ = 0+ with a polynomial approximation in the energy functional of the LogSE and then obtaining an energy regularized logarithmic Schrödinger equation (ERLogSE) via energy variation. Linear convergence is established between the solutions of ERLogSE and LogSE in terms of a small regularization parameter 0 < 𝜀 ≪ 1. Moreover, the conserved energy of the ERLogSE converges to that of LogSE quadratically, which significantly improves the linear convergence rate of the regularization method in the literature. Error estimates are also presented for solving the ERLogSE by using Lie–Trotter splitting integrators. Numerical results are reported to confirm our error estimates of the LER and of the time-splitting integrators for the ERLogSE. Finally, our results suggest that the LER performs better than regularizing the logarithmic nonlinearity in the LogSE directly.

中文翻译：

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对数薛定谔方程的局部能量正则化误差估计

对数非线性已在许多偏微分方程 (PDE) 中用于对各种应用中的问题进行建模。由于对数函数的奇异性，它给建立数学理论以及设计和分析具有这种非线性的偏微分方程的数值方法带来了巨大的困难。在这里，我们采用对数薛定谔方程（LogSE）作为原型模型。而不是正则化F(ρ) =ln ρ在 LogSE 中直接和全局地在文献中完成，我们通过首先正则化为 LogSE 提出局部能量正则化 (LER)F(ρ) = ρln ρ - ρ当地附近ρ = 0+在 LogSE 的能量泛函中进行多项式逼近，然后通过能量变化获得能量正则化对数薛定谔方程 (ERLogSE)。ERLogSE 和 LogSE 的解在一个小的正则化参数方面建立了线性收敛0 < 𝜀 ≪ 1. 此外，ERLogSE的守恒能量与LogSE二次收敛，显着提高了文献中正则化方法的线性收敛速度。还提出了使用 Lie-Trotter 分裂积分器求解 ERLogSE 的误差估计。报告的数值结果证实了我们对 ERLogSE 的 LER 和时间分割积分器的误差估计。最后，我们的结果表明，LER 比直接在 LogSE 中正则化对数非线性表现更好。