Pseudospectral Methods for Computing the Multiple Solutions of the Schrödinger Equation

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, combinedwith Legendre pseudospectralmethods. Then, starting from the non-trivial solution branches of the corresponding nonlinear problem, we further obtain the whole positive solution branch with D4 symmetry of the Schrödinger equation numerically by pseudo-arclength continuation algorithm. Next, we propose the extended systems, which can detect the fold and symmetrybreaking bifurcation points on the branch of the positive solutions with D4 symmetry. We also compute the multiple positive solutions with various symmetries of the Schrödinger equation by the branch switchingmethod 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. AMS subject classifications: 35Q55, 35J25, 37M20, 65M70