论文标题
具有单数非线性的非线性schrödinger方程的正则数字方法
Regularized numerical methods for the nonlinear Schrödinger equation with singular nonlinearity
论文作者
论文摘要
我们提出了具有单数非线性（SNLSE）的非线性Schrödinger方程的不同正规化和数值方法，包括正则化的Lie-Trotter时间分解时间（LTTS）方法和正常的Lawson-Lawson-type指数积分集成符（LTEI）方法。由于对单数非线性的爆炸，即，$ f（ρ）=ρ^α$带有固定指数$α<0 $时，当$ρ\ to 0^+$（$ρ\ to 0^+$（$ρ= |ψ|^2 $）表示密度代表着密度，$ψ$是$ψ$是复杂的波浪功能或订单的效率，并且可以使得精确地效果。 snlse。为了抑制圆形错误并避免接近$ρ= 0^+$的爆炸，提出了两种类型的SNLSE正规化类型的正规化参数$ 0 <\ eps \ ll 1 $。 One is based on the local energy regularization (LER) for the sNLSE via regularizing the energy density $F(ρ) = \frac{1}{α+1}ρ^{α+1}$ locally near $ρ= 0^+$ with a polynomial approximation and then obtaining a local energy regularized nonlinear Schrödinger equation via energy variation.另一个是全局非线性正则化，它直接将单数非线性$ f（ρ）=ρ^α$正规化，以避免爆炸接近$ρ= 0^+$。对于正规模型，我们将一阶Lie-Trotter时间拆分方法和Lawson-Type指数积分方法用于时间离散化，并将其与太空中的傅立叶假谱法相结合，以数值求解它们。提供了数值示例，以显示正则模型与SNLSE的收敛性，他们认为局部能量正则性能要比直接正规化全球奇异的非线性的性能要好。
We present different regularizations and numerical methods for the nonlinear Schrödinger equation with singular nonlinearity (sNLSE) including the regularized Lie-Trotter time-splitting (LTTS) methods and regularized Lawson-type exponential integrator (LTEI) methods. Due to the blowup of the singular nonlinearity, i.e., $f(ρ)=ρ^α$ with a fixed exponent $α<0$ goes to infinity when $ρ\to 0^+$ ($ρ= |ψ|^2$ represents the density with $ψ$ being the complex-valued wave function or order parameter), there are significant difficulties in designing accurate and efficient numerical schemes to solve the sNLSE. In order to suppress the round-off error and avoid blowup near $ρ= 0^+$, two types of regularizations for the sNLSE are proposed with a small regularization parameter $0 < \eps \ll 1$. One is based on the local energy regularization (LER) for the sNLSE via regularizing the energy density $F(ρ) = \frac{1}{α+1}ρ^{α+1}$ locally near $ρ= 0^+$ with a polynomial approximation and then obtaining a local energy regularized nonlinear Schrödinger equation via energy variation. The other one is the global nonlinearity regularization which directly regularizes the singular nonlinearity $f(ρ)=ρ^α$ to avoid blowup near $ρ= 0^+$. For the regularized models, we apply the first-order Lie-Trotter time-splitting method and Lawson-type exponential integrator method for temporal discretization and combine with the Fourier pseudospectral method in space to numerically solve them. Numerical examples are provided to show the convergence of the regularized models to the sNLSE and they suggest that the local energy regularization performs better than directly regularizing the singular nonlinearity globally.
