Pseudospectral Methods for Computing the Multiple Solutions of the Schrödinger Equation (Communications in Computational Physics, 2018)
In this paper, we first compute the multiple non-trivial solutions of theSchrödinger equation on a square, by using the Liapunov-Schmidt reduction andsymmetry-breaking bifurcation theory, combined with Legendre pseudospectral methods.Then, starting from the non-trivial solution branches of the corresponding nonlinearproblem, we further obtain the whole positive solution branch with $D_4$ symmetryof the Schrödinger equation numerically by pseudo-arclength continuation algorithm.Next, we propose the extended systems, which can detect the fold and symmetry-breakingbifurcation points on the branch of the positive solutions with $D_4$ symmetry.We also compute the multiple positive solutions with various symmetries of theSchrödinger equation by the branch switching method based on the Liapunov-Schmidtreduction. Finally, the bifurcation diagrams are constructed, showing the symmetry/peakbreaking phenomena of the Schrödinger equation. Numerical results demonstratethe effectiveness of these approaches.