Solution landscape of Onsager functional identifies non-axisymmetric critical points

We investigate the solution landscapes of the Onsager free-energy functional with different potential kernels, including the dipolar potential, the Maier--Saupe potential, the coupled dipolar/Maier--Saupe potential, and the Onsager potential. A uniform sampling method is implemented for the discretization of the Onsager functional, and the solution landscape of the Onsager functional is constructed by using the saddle dynamics coupled with downward/upward search algorithms. We first compute the solution landscapes with the dipolar and Maier--Saupe potentials, for which all critical points are axisymmetric. For the coupled dipolar/Maier--Saupe potential, the solution landscape shows a novel non-axisymmetric critical point, named as tennis, which exists for a wide range of parameters. We further demonstrate various non-axisymmetric critical points in the Onsager functional with the Onsager potential, including square, hexagon, octahedral, cubic, quarter, icosahedral, and dodecahedral states. The solution landscape provides an efficient approach to show the global structures as well as the bifurcations of critical points, which can not only verify the previous analytic results but also propose several conjectures based on the numerical findings.