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Bifurcation of nonlinear bound states in the periodic Gross-Pitaevskii equation with 𝒫𝒯-symmetry

Part of: Spectral theory and eigenvalue problems Equations and inequalities involving nonlinear operators General mathematical topics and methods in quantum theory

Published online by Cambridge University Press:  24 January 2019

Tomáš Dohnal  and
Dmitry Pelinovsky
Tomáš Dohnal
Affiliation:
Fachbereich Mathematik Technical University Dortmund, Vogelpothsweg 87, Dortmund 44221, Germany (tomas.dohnal@math.tu-dortmund.de)
Dmitry Pelinovsky
Affiliation:
Department of Mathematics, McMaster University, Hamilton, OntarioL8S 4K1, Canada (dmpeli@math.mcmaster.ca) and Department of Applied Mathematics, Nizhny Novgorod State Technical University, 24 Minin street, Nizhny Novgorod603950, Russia
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Abstract

The stationary Gross–Pitaevskii equation in one dimension is considered with a complex periodic potential satisfying the conditions of the 𝒫𝒯 (parity-time reversal) symmetry. Under rather general assumptions on the potentials, we prove bifurcations of 𝒫𝒯-symmetric nonlinear bound states from the end points of a real interval in the spectrum of the non-selfadjoint linear Schrödinger operator with a complex 𝒫𝒯-symmetric periodic potential. The nonlinear bound states are approximated by the effective amplitude equation, which bears the form of the cubic nonlinear Schrödinger equation. In addition, we provide sufficient conditions for the appearance of complex spectral bands when the complex 𝒫𝒯-symmetric potential has an asymptotically small imaginary part.

Keywords

bifurcationnonlinear bound statesspectral intervalsnon-selfadjoint linear Schrödinger operator𝒫𝒯-symmetry

MSC classification

Primary: 47J10: Nonlinear spectral theory, nonlinear eigenvalue problems
Secondary: 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory 81Q12: Non-selfadjoint operator theory in quantum theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 1 , February 2020 , pp. 171 - 204
Copyright
Copyright © Royal Society of Edinburgh 2019

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