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The fractional nonlinear [Formula: see text] dimer.

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This study explores fractional discrete nonlinear Schrodinger dimers. Fractional derivatives decrease oscillatory exchange amplitude, but nonlinearity can induce self-trapping, unlike standard dimers.

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Area of Science:

  • Nonlinear dynamics
  • Fractional calculus
  • Quantum mechanics

Background:

  • Investigating nonlinear Schrodinger equations in dimer systems.
  • Exploring the impact of fractional time derivatives on system dynamics.
  • Utilizing parity-time ([Formula: see text]) symmetry in discrete systems.

Purpose of the Study:

  • Examine the dynamics of a fractional discrete nonlinear Schrodinger dimer.
  • Analyze the exchange dynamics for localized initial conditions.
  • Understand the role of fractional derivatives and nonlinearity on system behavior.

Main Methods:

  • Laplace transformation for solving the linear fractional dimer.
  • Direct explicit Grunwald algorithm for numerical computation in the nonlinear regime.
  • Analysis of localized initial conditions to study exchange dynamics.

Main Results:

  • Fractional derivatives lead to a decreasing time envelope for oscillatory exchange.
  • Parity-time ([Formula: see text]) symmetry amplifies oscillations below a threshold, causing unbounded growth beyond it.
  • Nonlinearity can arrest unbounded growth, leading to self-trapped states, with decreasing trapped fraction as nonlinearity increases.

Conclusions:

  • Fractional derivatives significantly alter energy exchange dynamics in nonlinear dimers.
  • The interplay between fractional calculus, [Formula: see text] symmetry, and nonlinearity dictates system stability and localization.
  • Nonlinearity's effect on self-trapping in fractional dimers contrasts with the standard non-fractional case.