Xiaonan Wu

Jiwei Zhang, Zhenli Xu, Xiaonan Wu

An efficient method is proposed for numerical solutions of nonlinear Schrödinger equations in an unbounded domain. Through approximating the kinetic energy term by a one-way equation and uniting it with the potential energy equation, absorbing boundary conditions are designed to truncate the unbounded domain, which are in nonlinear form and can perfectly absorb the waves outgoing from the truncated domain. We examine the stability of the induced initial boundary value problems defined on the computational domain with the boundary conditions by a normal mode analysis. Numerical examples are given to illustrate the stable and tractable advantages of the method.

Zhenli Xu, Houde Han, Xiaonan Wu

We propose an adaptive approach in picking the wave-number parameter of absorbing boundary conditions for Schrödinger-type equations. Based on the Gabor transform which captures local frequency information in the vicinity of artificial boundaries, the parameter is determined by an energy-weighted method and yields a quasi-optimal absorbing boundary conditions. It is shown that this approach can minimize reflected waves even when the wave function is composed of waves with different group velocities. We also extend the split local absorbing boundary (SLAB) method [Z. Xu and H. Han, {\it Phys. Rev. E}, 74(2006), pp. 037704] to problems in multidimensional nonlinear cases by coupling the adaptive approach. Numerical examples of nonlinear Schrödinger equations in one- and two dimensions are presented to demonstrate the properties of the discussed absorbing boundary conditions.