Qinglin Tang

Numerical methods and comparison for computing dark and bright solitons in the nonlinear Schrödinger equation

In this paper, we propose new efficient and accurate numerical methods for computing dark solitons and review some existing numerical methods for bright and/or dark solitons in the nonlinear Schrodinger equation (NLSE), and compare them numerically in terms of accuracy and efficiency. We begin with a review of dark and bright solitons of NLSE with defocusing and focusing cubic nonlinearities, respectively. For computing dark solitons, to overcome the nonzero and/or non-rest (or highly oscillatory) phase background at far field, we design efficient and accurate numerical methods based on accurate and simple artificial boundary conditions or a proper transformation to rest the highly oscillatory phase background. Stability and conservation laws of these numerical methods are analyzed. For computing interactions between dark and bright solitons, we compare the efficiency and accuracy of the above numerical methods and different existing numerical methods for computing bright solitons of NLSE, and identify the most efficient and accurate numerical methods for computing dark and bright solitons as well as their interactions in NLSE. These numerical methods are applied to study numerically the stability and interactions of dark and bright solitons in NLSE. Finally, they are extended to solve NLSE with general nonlinearity and/or external potential and coupled NLSEs with vector solitons.

A UNIFORMLY ACCURATE MULTISCALE TIME INTEGRATOR PSEUDOSPECTRAL METHOD FOR THE DIRAC EQUATION IN THE NONRELATIVISTIC LIMIT REGIME

We propose and rigourously analyze a multiscale time integrator Fourier pseudospectral (MTI-FP) method for the Dirac equation with a dimensionless parameter

A Simple and Efficient Numerical Method for Computing the Dynamics of Rotating Bose--Einstein Condensates via Rotating Lagrangian Coordinates

\varepsilon\in(0,1]

Accurate and efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates via the nonuniform FFT

which is inversely proportional to the speed of light. In the nonrelativistic limit regime, i.e.

On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions

0<\varepsilon\ll 1

Computing the ground state and dynamics of the nonlinear Schrödinger equation with nonlocal interactions via the nonuniform FFT

, the solution exhibits highly oscillatory propagating waves with wavelength

An efficient spectral method for computing dynamics of rotating two-component Bose-Einstein condensates via coordinate transformation

O(\varepsilon^2)

Numerical Methods and Comparison for the Dirac Equation in the Nonrelativistic Limit Regime

and

A variational-difference numerical method for designing progressive-addition lenses

O(1)

The Numerical Study of the Ground States of Spin-1 Bose-Einstein Condensates with Spin-Orbit-Coupling

in time and space, respectively. Due to the rapid temporal oscillation, it is quite challenging in designing and analyzing numerical methods with uniform error bounds in