Bifurcation of nonlinear eigenvalues in problems with antilinear symmetry

TL;DR

This paper investigates how nonlinear eigenvalues with antilinear symmetry, such as PT-symmetry, bifurcate from real linear eigenvalues, ensuring they remain real and symmetric in various physical models.

Contribution

It establishes that nonlinear eigenvalues bifurcating from real eigenvalues under antilinear symmetry stay real and symmetric, with applications to multiple physical systems.

Findings

Nonlinear eigenvalues bifurcating from real eigenvalues remain real.

Nonlinear eigenfunctions preserve symmetry under antilinear symmetry.

Numerical analysis confirms theoretical results in physical models.

Abstract

Many physical systems can be described by nonlinear eigenvalues and bifurcation problems with a linear part that is non-selfadjoint e.g. due to the presence of loss and gain. The balance of these effects is reflected in an antilinear symmetry, like e.g. the PT-symmetry, of the problem. Under this condition we show that the nonlinear eigenvalues bifurcating from real linear eigenvalues remain real and the corresponding nonlinear eigenfunctions remain symmetric. The abstract results are applied in a number of physical models of Bose-Einstein condensation, nonlinear optics and superconductivity, and further numerical analysis is performed.