Homotopy Methods for Eigenvector-Dependent Nonlinear Eigenvalue Problems

TL;DR

This paper develops homotopy methods to solve eigenvector-dependent nonlinear eigenvalue problems from discretized Gross-Pitaevskii equations, proving existence, uniqueness, and regularity of solution paths, with numerical verification.

Contribution

It introduces homotopy techniques tailored for eigenvector-dependent nonlinear eigenvalue problems, establishing theoretical properties and providing numerical validation.

Findings

Existence and uniqueness of positive eigenvectors proved.

Homotopy paths are regular and bounded.

Numerical results confirm theoretical predictions.

Abstract

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.