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Bifurcation of nonlinear bound states in the periodic Gross-Pitaevskii equation with 𝒫𝒯-symmetry
Published online by Cambridge University Press: 24 January 2019
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.
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- 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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