7 research outputs found
Adaptive absorbing boundary conditions for Schrodinger-type equations: application to nonlinear and multi-dimensional problems
We propose an adaptive approach in picking the wave-number parameter of
absorbing boundary conditions for Schr\"{o}dinger-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 Schr\"{o}dinger
equations in one- and two dimensions are presented to demonstrate the
properties of the discussed absorbing boundary conditions.Comment: 18 pages; 12 figures. A short movie for the 2D NLS equation with
absorbing boundary conditions can be downloaded at
http://home.ustc.edu.cn/~xuzl/movie.avi. To appear in Journal of
Computational Physic
Bifurcation analysis and exact solutions for a class of generalized time-space fractional nonlinear Schrödinger equations
In this work, we focus on a class of generalized time-space fractional nonlinear Schrödinger equations arising in mathematical physics. After utilizing the general mapping deformation method and theory of planar dynamical systems with the aid of symbolic computation, abundant new exact complex doubly periodic solutions, solitary wave solutions and rational function solutions are obtained. Some of them are found for the first time and can be degenerated to trigonometric function solutions. Furthermore, by applying the bifurcation theory method, the periodic wave solutions and traveling wave solutions with the corresponding phase orbits are easily obtained. Moreover, some numerical simulations of these solutions are portrayed, showing the novelty and visibility of the dynamical structure and propagation behavior of this model