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 - <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>