Numerical Methods for the Nonlinear Schrödinger Equation with Nonzero Far-field Conditions

Abstract

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.