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Fractional nonlinear electrical lattice.

Mario I Molina1

  • 1Departamento de Física, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile.

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Summary
This summary is machine-generated.

This study explores fractional electrical lattices, revealing that decreasing the fractional exponent leads to ballistic behavior and fewer electrical discrete solitons. The system exhibits degeneracy as the exponent approaches zero.

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Area of Science:

  • Condensed Matter Physics
  • Nonlinear Dynamics
  • Fractional Calculus

Background:

  • Investigates one-dimensional nonlinear electrical lattices.
  • Replaces the standard discrete Laplacian with a fractional discrete Laplacian.
  • Introduces long-range intersite coupling with power-law decay.

Purpose of the Study:

  • To analyze the linear and nonlinear modes of fractional electrical lattices.
  • To understand the impact of the fractional exponent on system dynamics.
  • To compute plane wave spectra, mean-square displacement, and nonlinear modes.

Main Methods:

  • Analytical computation of plane wave spectra and mean-square displacement (MSD).
  • Closed-form solutions derived using regularized hypergeometric functions.
  • Numerical computation of low-lying nonlinear modes and modulational stability analysis.

Main Results:

  • MSD exhibits ballistic behavior (MSD∼t²) at long times for all fractional exponents.
  • Decreasing the fractional exponent reduces bandwidth and increases state degeneracy.
  • Fewer electrical discrete solitons are generated as the fractional exponent decreases, eventually collapsing into one.

Conclusions:

  • Fractionalization of the discrete Laplacian significantly alters lattice dynamics.
  • The system transitions towards degeneracy and reduced soliton formation with decreasing fractional exponent.
  • Fractional electrical lattices offer a tunable platform for exploring nonlinear phenomena.