A hyperplane-constrained continuation method for near singularity in coupled nonlinear Schrödinger equations

The continuation method is a useful numerical tool for solving differential equations to obtain multiform solutions and allow bifurcation analysis. However, when a standard continuation method is used to solve a type of time-independent m-coupled nonlinear Schrödinger (NLS) equations that can be used to model nonlinear optics, nearly singular systems arise in the computations of prediction and correction search directions and detections of bifurcations. To overcome the stability and efficiency problems that exist in standard continuation methods, we propose a new hyperplane-constrained continuation method by adding additional constraints to prevent the singularities while tracking the solution curves. Aimed at the 3-coupled DNLS equations, we conduct theoretical analysis to the solutions and bifurcations on the primal stalk solution curve. The proposed algorithms have been implemented successfully to demonstrate numerical solution profiles, energies, and bifurcation diagrams in various settings.