Computational methods for the dynamics of the nonlinear Schroedinger/Gross-Pitaevskii equations
In this talk, I begin wtih the (nonlinear) Schroedinger/Gross-Pitaevskii equations (NLSE/GPE) for modeling Bose-Einstein condensation (BEC), nonlinear optics, quantum physics and chemistry, etc., and review some dynamical properties of NLSE/GPE including conserved quantities, dispersion relation,
center-of-mass dynamics, soliton solutions and semiclassical limits.
Different numerical methods will be presented including finite difference time domain (FDTD) methods and time-splitting spectral (TSSP) method, and their error estimates and comparison will be discussed. Extensions to NLSE/GPE with an angular momentum rotation term, non-local dipole-dipole interaction, multi-component and logarithmic nonlinearity will be presented.
Applications to soliton interactions, collapse and explosion of BEC, quantum transport and quantized vortex interaction will be investigated.
Finally, I will report our very recent results of energy regularization methods for NLSE with logarithmic nonlinearity and improved uniform error bounds
on TSSP for the long-time dynamics of NLSE by using the regularization compensation oscillatory (RCO) technique.