Numerical solutions to an Allen-Cahn type equation

Our research focuses on second order nonlinear elliptic partial differential equations, specifically an Allen-Cahn type problem. Using variational methods, we locate known numerical solutions to this equation by finding minima of a related functional. This analysis utilizes a steepest descent program. Since these solutions are distinct, we may employ the Mountain Pass Algorithm, introduced by Choi and McKenna in 1993, to find additional solutions corresponding to saddle points of the functional. We then investigate the qualitative properties of these solutions.