  We present different regularizations and numerical methods for the nonlinear
Schr\"odinger 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., $f(\rho)=\rho^{\alpha}$ with a fixed exponent
$\alpha<0$ goes to infinity when $\rho \to 0^+$ ($\rho = |\psi|^2$ represents
the density with $\psi$ 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 $\rho = 0^+$, 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 $F(\rho) =
\frac{1}{\alpha+1}\rho^{\alpha+1}$ locally near $\rho = 0^+$ with a polynomial
approximation and then obtaining a local energy regularized nonlinear
Schr\"odinger equation via energy variation. The other one is the global
nonlinearity regularization which directly regularizes the singular
nonlinearity $f(\rho)=\rho^{\alpha}$ to avoid blowup near $\rho = 0^+$. 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.

Numerical Analysis;Numerical Analysis