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Mixed Fourier-Legendre Spectral Methods for the Multiple Solutions of the Schrodinger Equation on the Unit Disk

    Zhao-Xiang Li Affiliation
    ; Ji Lao Affiliation
    ; Zhong-Qing Wang Affiliation

Abstract

In this paper, we first compute the multiple non-trivial solutions of the Schrodinger equation on a unit disk, by using the Liapunov-Schmidt reduction and symmetry-breaking bifurcation theory, combined with the mixed Fourier-Legendre spectral and pseudospectral methods. After that, we propose the extended systems, which can detect the symmetry-breaking bifurcation points on the branch of the O(2) symmetric positive solutions. We also compute the multiple positive solutions with various symmetries of the Schrodinger 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 Schr¨odinger equation. Numerical results demonstrate the effectiveness of these approaches.

Keyword : approximation algorithm, bifurcation diagrams, multiple solutions, computational experiment, positive solution

How to Cite
Li, Z.-X., Lao, J., & Wang, Z.-Q. (2017). Mixed Fourier-Legendre Spectral Methods for the Multiple Solutions of the Schrodinger Equation on the Unit Disk. Mathematical Modelling and Analysis, 22(2), 167-185. https://doi.org/10.3846/13926292.2017.1285362
Published in Issue
Mar 18, 2017
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This work is licensed under a Creative Commons Attribution 4.0 International License.