Pseudospectral Methods for Computing the Multiple Solutions of the Schrodinger Equation
COMMUNICATIONS IN COMPUTATIONAL PHYSICS(2018)
摘要
In this paper, we first compute the multiple non-trivial solutions of the Schrodinger 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 Schrodinger 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 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 Schrodinger equation. Numerical results demonstrate the effectiveness of these approaches.
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关键词
Schrodinger equation,multiple solutions,symmetry-breaking bifurcation theory,Liapunov-Schmidt reduction,pseudospectral method
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