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

• Published in: Mathematical modelling and analysis Vol. 22; no. 2; pp. 167 - 185
• Main Authors: Li, Zhao-Xiang, Laob, Ji, Wang, Zhong-Qing
• Format: Journal Article
• Language: English
• Published: Taylor & Francis 04.03.2017; Vilnius Gediminas Technical University
• Subjects: approximation algorithm; bifurcation diagrams; Bifurcation theory; Bifurcations; computational experiment; Disks; Mathematical models; Mathematical research; multiple solutions; positive solution; Reduction; Schroedinger equation; Schrödinger equation; Spectra; Spectral decomposition (Mathematics); Spectral methods; Symmetry; approximation algorithm; positive solution; multiple solutions; computational experiment; bifurcation diagrams
• ISSN: 1392-6292; 1648-3510
• DOI: 10.3846/13926292.2017.1285362

In this paper, we first compute the multiple non-trivial solutions of the Schrödinger 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 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.