A numerical method for finding multiple co-existing solutions to nonlinear cooperative systems

In this paper, a local min-orthogonal method is developed to solve cooperative nonlinear elliptic systems for multiple co-existing solutions. A characterization of co-existing critical points of a dual functional is established and used as a mathematical justification for the method. The method is then implemented to numerically solve two coupled nonlinear Schrödinger equations which model spatial vector solitons propagating in a saturable bulk nonlinear medium for multiple co-existing solutions.