In this paper we present numerical methods
for the nonlinear Schrödinger equations (NLS)
in the semiclassical regimes:
\[i \vep\; u_t^\vep=-\fl{\vep^2}{2}\;\btu u^\vep+V(\bx)u^\vep+
f(|u^\vep|^2)u^\vep, \qquad \bx\in{\Bbb R}^d,\]
with nonzero far-field conditions.
A time-splitting cosine-spectral (TS-Cosine) method is presented when
the nonzero far-field conditions are or can be reduced to homogeneous
Neumann conditions, a time-splitting Chebyshev-spectral (TS-Chebyshev)
method is proposed for more general nonzero far-field conditions,
and an efficient and accurate numerical method in which
we use polar coordinates to properly match the nonzero
far-field conditions is presented for computing
dynamics of quantized vortex lattice of NLS in two dimensions (2D).
All the methods are explicit, unconditionally stable and time reversible.
Furthermore, TS-Cosine is time-transverse invariant and
conserves the position density,
where TS-Chebyshev can deal with more general
nonzero far-field conditions.
Extensive numerical tests are presented for
linear constant/harmonic oscillator potential, defocusing
nonlinearity of NLS to study the $\vep$-resolution of the methods.
Our numerical tests suggest the following `optimal' $\vep$-resolution
of the methods for
obtaining `correct'
physical observables in the semi-classical regimes:
time step $k$-independent of $\vep$ and mesh size $h=O(\vep)$ for
linear case; $k=O(\vep)$ and $h=O(\vep)$ for defocusing
nonlinear case. The methods
are applied to study numerically the semiclassical limits of NLS
in 1D and the dynamics of quantized vortex lattice of NLS in 2D
with nonzero far-field conditions.