Pseudospectral methods for computing the multiple solutions of the Lane-Emden equation

Based on the Liapunov-Schmidt reduction and symmetry-breaking bifurcation theory, we compute and visualize multiple solutions of the Lane-Emden equation on a square and a disc, using Legendre and Fourier-Legendre pseudospectral methods. Starting from the nontrivial solution branches of the corresponding nonlinear bifurcation problem, we obtain multiple solutions of Lane-Emden equation with various symmetries numerically. Numerical results demonstrate the effectiveness of these approaches.