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.