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Symmetry 2017, 9(8), 165; https://doi.org/10.3390/sym9080165

Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear PT-Symmetry to the Nonlinear Coupling of Channels

1
Nuclear Physics Institute of the CAS, Hlavní 130, 250 68 Řež, Czech Republic
2
Institute of Systems Science, Durban University of Technology, Durban 4000, South Africa
*
Author to whom correspondence should be addressed.
Received: 20 June 2017 / Revised: 31 July 2017 / Accepted: 8 August 2017 / Published: 18 August 2017
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Abstract

Schrödinger equations with non-Hermitian, but PT -symmetric quantum potentials V ( x ) found, recently, a new field of applicability in classical optics. The potential acquired there a new physical role of an “anomalous” refraction index. This turned attention to the nonlinear Schrödinger equations in which the interaction term becomes state-dependent, V ( x ) W ( ψ ( x ) , x ) . Here, the state-dependence in W ( ψ ( x ) , x ) is assumed logarithmic, and some of the necessary mathematical assumptions, as well as some of the potential phenomenological consequences of this choice are described. Firstly, an elementary single-channel version of the nonlinear logarithmic model is outlined in which the complex self-interaction W ( ψ ( x ) , x ) is regularized via a deformation of the real line of x into a self-consistently constructed complex contour C. The new role played by PT -symmetry is revealed. Secondly, the regularization is sought for a multiplet of equations, coupled via the same nonlinear self-interaction coupling of channels. The resulting mathematical structures are shown to extend the existing range of physics covered by the logarithmic Schrödinger equations. View Full-Text
Keywords: PT symmetry; nonlinear Schrödinger equations; logarithmic nonlinearities; coupled-channel systems; regularizations PT symmetry; nonlinear Schrödinger equations; logarithmic nonlinearities; coupled-channel systems; regularizations
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Znojil, M.; Růžička, F.; Zloshchastiev, K.G. Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear PT-Symmetry to the Nonlinear Coupling of Channels. Symmetry 2017, 9, 165.

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