Volume 23, Issue 3
Pseudospectral Methods for Computing the Multiple Solutions of the Schrödinger Equation

Commun. Comput. Phys., 23 (2018), pp. 822-845.

Published online: 2018-03

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• 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 $D_4$ 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 $D_4$ 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.

• Keywords

Schrödinger equation, multiple solutions, symmetry-breaking bifurcation theory, Liapunov-Schmidt reduction, pseudospectral method.

35Q55, 35J25, 37M20, 65M70

• BibTex
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@Article{CiCP-23-822, author = {}, title = {Pseudospectral Methods for Computing the Multiple Solutions of the Schrödinger Equation}, journal = {Communications in Computational Physics}, year = {2018}, volume = {23}, number = {3}, pages = {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 $D_4$ 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 $D_4$ 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2017-0020}, url = {http://global-sci.org/intro/article_detail/cicp/10550.html} }
TY - JOUR T1 - Pseudospectral Methods for Computing the Multiple Solutions of the Schrödinger Equation JO - Communications in Computational Physics VL - 3 SP - 822 EP - 845 PY - 2018 DA - 2018/03 SN - 23 DO - http://doi.org/10.4208/cicp.OA-2017-0020 UR - https://global-sci.org/intro/article_detail/cicp/10550.html KW - Schrödinger equation, multiple solutions, symmetry-breaking bifurcation theory, Liapunov-Schmidt reduction, pseudospectral method. AB -

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 $D_4$ 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 $D_4$ 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.

Zhao-Xiang Li, Ji Lao & Zhong-Qing Wang. (2020). Pseudospectral Methods for Computing the Multiple Solutions of the Schrödinger Equation. Communications in Computational Physics. 23 (3). 822-845. doi:10.4208/cicp.OA-2017-0020
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