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Pseudospectral Methods for Computing the Multiple Solutions of the Schrödinger Equation

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