Bifurcation of nonlinear eigenvalues in problems with antilinear symmetry

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.

arXiv comment30 pages, 11 figures, added examples

References in corpus (11)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Visualization of Branch Points in PT-Symmetric Waveguides
• On the reality of the eigenvalues for a class of PT-symmetric oscillators
• Bifurcation diagram and pattern formation in superconducting wires with electric currents
• Pseudospectra in non-Hermitian quantum mechanics
• Closed formula for the metric in the Hilbert space of a PT-symmetric model
• Root system of singular perturbations of the harmonic oscillator type operators
• Dipolar Bose-Einstein condensates in a PT-symmetric double-well potential
• Differential operators admitting various rates of spectral projection growth
• Riesz Bases for p-Subordinate Perturbations of Normal Operators
• Spectral projections of the complex cubic oscillator

Cited by in corpus (2)

• Nonlinear waves in $\mathcal{P}\mathcal{T}$-symmetric systems
• Small-amplitude nonlinear modes under the combined effect of the parabolic potential, nonlocality and $\mathcal{P}\mathcal{T}$ symmetry