Pseudospectral Methods for Computing the Multiple Solutions of the Schrödinger Equation
Year: 2018
Author: Zhao-Xiang Li, Ji Lao, Zhong-Qing Wang
Communications in Computational Physics, Vol. 23 (2018), Iss. 3 : pp. 822–845
Abstract
In this paper, we first compute the multiple non-trivial solutions of the Schrödinger equation on a square, by using the Liapunov-Schmidt reduction and symmetry-breaking bifurcation theory, combined with Legendre pseudospectral methods. Then, starting from the non-trivial solution branches of the corresponding nonlinear problem, we further obtain the whole positive solution branch with D4 symmetry of the Schrödinger equation numerically by pseudo-arclength continuation algorithm. Next, we propose the extended systems, which can detect the fold and symmetry-breaking bifurcation points on the branch of the positive solutions with D4 symmetry. We also compute the multiple positive solutions with various symmetries of the Schrödinger equation by the branch switching method based on the Liapunov-Schmidt reduction. Finally, the bifurcation diagrams are constructed, showing the symmetry/peak breaking phenomena of the Schrödinger equation. Numerical results demonstrate the effectiveness of these approaches.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2017-0020
Communications in Computational Physics, Vol. 23 (2018), Iss. 3 : pp. 822–845
Published online: 2018-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 24
Keywords: Schrödinger equation multiple solutions symmetry-breaking bifurcation theory Liapunov-Schmidt reduction pseudospectral method.
Author Details
Zhao-Xiang Li Email
Ji Lao Email
Zhong-Qing Wang Email