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J Fujioka1, A Espinosa1, R F Rodríguez1

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Researchers explored fractional extensions of the nonlinear Schrödinger (NLS) equation. A new fractional NLS equation with a temporal fractional derivative (TFD) of order α < 2 describes radiating solitons and conserves key physical quantities.

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Area of Science:

  • Nonlinear Physics
  • Soliton Dynamics
  • Fractional Calculus

Background:

  • Previous work established fractional nonlinear Schrödinger (NLS) equations with temporal fractional derivatives (TFDs) of order α > 2 for solitary wave propagation.
  • The behavior of solitons under different fractional derivative orders was not fully understood.

Purpose of the Study:

  • To investigate a novel fractional extension of the NLS equation involving a TFD of order α < 2.
  • To analyze the propagation characteristics of solitons described by this new fractional NLS equation.
  • To explore the underlying physical mechanisms and conservation laws associated with this model.

Main Methods:

  • Development of a fractional nonlinear Schrödinger (NLS) equation with a temporal fractional derivative (TFD) of order α < 2.
  • Analysis of radiating soliton propagation within this fractional NLS framework.
  • Investigation of resonances between pulses and linear modes to explain radiation emission.
  • Derivation of the fractional NLS equation from a Lagrangian density.
  • Application of a fractional Noether's theorem to identify conserved quantities.

Main Results:

  • The new fractional NLS equation with α < 2 accurately describes the propagation of radiating solitons.
  • Radiation emission is attributed to frequency resonances between the solitons and the system's linear modes.
  • The fractional NLS equation is derivable from a Lagrangian.
  • Conservation of Hamiltonian, momentum, and energy is predicted by the fractional Noether's theorem.

Conclusions:

  • A new fractional nonlinear Schrödinger (NLS) equation with a temporal fractional derivative (TFD) of order α < 2 governs radiating soliton propagation.
  • Resonance phenomena explain the radiation emitted by these solitons.
  • Fundamental conservation laws (Hamiltonian, momentum, energy) are preserved in this fractional system, as indicated by the fractional Noether's theorem.