@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 = {<p style="text-align: justify;">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.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2017-0020},
url = {http://global-sci.org/intro/article_detail/cicp/10550.html}
}