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Discrete solitons dynamics in \(\mathscr {PT}\)-symmetric oligomers with complex-valued couplings

A Correction to this article was published on 24 June 2021

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Abstract

We consider an array of double oligomers in an optical waveguide device. A mathematical model for the system is the coupled discrete nonlinear Schrödinger equations, where the gain-and-loss parameter contributes to the complex-valued linear coupling. The array caters to an optical simulation of the parity-time (\(\mathscr {PT}\))-symmetry property between the coupled arms. The system admits fundamental bright discrete soliton solutions. We investigate their existence and spectral stability using perturbation theory analysis. These analytical findings are verified further numerically using the Newton–Raphson method and a standard eigenvalue-problem solver. Our study focuses on two natural discrete modes of the solitons: single- and double-excited-sites, also known as onsite and intersite modes, respectively. Each of these modes acquires three distinct configurations between the dimer arms, i.e., symmetric, asymmetric, and antisymmetric. Although both intersite and onsite discrete solitons are generally unstable, the latter can be stable, depending on the combined values of the propagation constant, horizontal linear coupling coefficient, and gain–loss parameter.

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Acknowledgements

We are grateful to Professor Hadi Susanto, Department of Mathematical Sciences, University of Essex, UK and Department of Mathematics, Khalifa University, Abu Dhabi, The United Arab Emirates, for his assistance and valuable comments in improving this paper significantly.

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Correspondence to N. Karjanto.

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Kirikchi, O.B., Karjanto, N. Discrete solitons dynamics in \(\mathscr {PT}\)-symmetric oligomers with complex-valued couplings. Nonlinear Dyn 103, 2769–2782 (2021). https://doi.org/10.1007/s11071-021-06217-5

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Keywords

  • Dimer and oligomer
  • \(\mathscr {PT}\)-symmetry
  • Discrete NLS equation
  • Bright soliton
  • Onsite and intersite modes
  • Dimer arm configuration

Mathematics Subject Classification

  • 74J30
  • 75J35
  • 78A40
  • 78A60
  • 78M35
  • 35Q55
  • 37K40
  • 37K45
  • 35C08