An orthogonal subspace minimization method for finding multiple solutions to the defocusing nonlinear Schr?dinger equation with symmetry

摘要

An orthogonal subspace minimization method is developed for finding multiple (eigen) solutions to the defocusing nonlinear Schr?dinger equation with symmetry. As such solutions are unstable, gradient search algorithms are very sensitive to numerical errors, will easily break symmetry, and will lead to unwanted solutions. Instead of enforcing a symmetry by the Haar projection, the authors use the knowledge of previously found solutions to build a support for the minimization search. With this support, numerical errors can be partitioned into two components, sensitive versus insensitive to the negative gradient search. Only the sensitive part is removed by an orthogonal projection. Analysis and numerical examples are presented to illustrate the method. Numerical solutions with some interesting phenomena are captured and visualized by their solution profile and contour plots.