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

Zhao-Xiang Li, Ji Lao & Zhong-Qing Wang

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 D4 symmetry of the Schrödinger equation numerically by pseudo-arclength continuation algorithm. Next, we propose the extended systems, which can detect the fold and symmetrybreaking 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.

  • Keywords

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

  • AMS Subject Headings

35Q55, 35J25, 37M20, 65M70

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COPYRIGHT: © Global Science Press

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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 D4 symmetry of the Schrödinger equation numerically by pseudo-arclength continuation algorithm. Next, we propose the extended systems, which can detect the fold and symmetrybreaking 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.

}, 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://dor.org/10.4208/cicp.OA-2017-0020 UR - https://global-sci.org/intro/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 D4 symmetry of the Schrödinger equation numerically by pseudo-arclength continuation algorithm. Next, we propose the extended systems, which can detect the fold and symmetrybreaking 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.

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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