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

Abstract

The second one reads [16, 17] u ε (x, t) → A ∞ exp i x · S ∞ ε |x| , as |x| → ∞, (1.4) where S ∞ = (S 1∞ , · · · , S d∞) T with A ∞ , S 1∞ , · · · , S d∞ constants, |x| = x 2 1 + · · · + x 2 d , x · S ∞ = x 1 S 1∞ + · · · + x d S d∞ , and the third one which is usually used for studying vortex motion of NLS [23, 28, 10] in 2D is u ε (x 1 , x 2 , t) → A ∞ exp (i mθ) , as r = |x| = x 2 1 + x 2 2 → ∞, (1.5) here (r, θ) is the polar coordinate, A ∞ is a constant and m is an integer. For the boundedness of the energy functional of the NLS (1.1) [21], we assume [V (x) + f (A 2 ∞)]A ∞ = 0, as |x| → ∞. (1.6) In quantum mechanics, the wave function is an auxiliary quantity used to compute the primary physical quantities such as the position density ρ ε (x, t) = |u ε (x, t)| 2 (1.7) and the current density J ε (x, t) = ε Im(u ε (x, t) ∇u ε (x, t)) = ε 2i (u ε ∇u ε − u ε ∇u ε), (1.8) where "-" denotes complex conjugation. The general form of (1.1) covers many nonlinear Schrödinger equations arising in various different applications. For example, when f (ρ) ≡ 0, A ∞ = 0 in (1.4), it reduces to the linear Schrödinger equation; when V (x) ≡ 0, f (ρ) = β ε ρ, A ∞ = 0.0 in (1.4), it is the cubic NLS (called the focusing NLS if β ε < 0 and the defocusing NLS if β ε > 0 [21]); when V (x) ≡ 0, f (ρ) = ρ − 1, A ∞ = 1.0 in (1.4), it corresponds to the propagation of a wave beam in a defocusing medium [32, 33]; when V (x) ≡ 0, f (ρ) = ρ − 1, A ∞ = 1.0 in (1.5), it corresponds to a vortex motion of NLS in 2D [28, 23, 10]. It is well known that the equation (1.1) propagates oscillations in space and time, preventing u ε from converging strongly as ε → 0. Much progress has been made recently in analytical understanding semiclassical limits of the linear Schödinger equation (i.e. f (ρ) ≡ 0 in (1.1)), particularly by the introduction of tools from mi-crolocal analysis, such as defect measures [13], H-measures [30], and Wigner measures [12, 14, 24]. These techniques have not been successfully extended to the semiclassical limit of the NLS, except that the 1D defocusing (cubically) NLS (1.1) was solved by using techniques of inverse scattering [16, 17]. Thus it is a very interesting problem to study the semiclassical limit of NLS numerically. The oscillatory nature of solutions of the Schrödinger equation with small ε provides severe numerical burdens. In [25, 26], Markowich et al. studied the finite difference approximation of the linear Schrödinger equation with small ε and zero far-field condition. Their results show that, for the best combination of the time and space discretizations, one needs the following constraints in order to guarantee good approximations to all (smooth) observables for ε small [25, 26]: mesh size h = o(ε) and time step k = o(ε). The same or more severe meshing constraint is required by the finite difference approximation of the NLS in the semi-classical regime with zero