Pseudospectral Methods for Computing the Multiple Solutions of the Schrödinger Equation

Pseudospectral Methods for Computing the Multiple Solutions of the Schrödinger Equation

Year:    2018

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 $D_4$ 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.

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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.