Homotopy Methods for Eigenvector-Dependent Nonlinear Eigenvalue Problems

Eigenvector-dependent nonlinear eigenvalue problems are considered which arise from the finite difference discretizations of the Gross-Pitaevskii equation. Existence and uniqueness of positive eigenvector for both one and two dimensional cases and existence of antisymmetric eigenvector for one dimensional case are proved. In order to compute eigenpairs corresponding to excited states as well as ground state, homotopies for both one and two dimensional problems are constructed respectively and the homotopy paths are proved to be regular and bounded. Numerical results are presented to verify the theories derived for both one and two dimensional problems.