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Basic fractional nonlinear-wave models and solitons.

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Summary
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This review explores wave propagation models in fractional media, detailing Riesz fractional derivatives and soliton behaviors. Experimental realizations of fractional group-velocity dispersion are also summarized.

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Area of Science:

  • Physics
  • Optics
  • Quantum Mechanics

Background:

  • Fractional media models are crucial for understanding wave propagation.
  • Laskin's fractional quantum mechanics and optical setups inspire these models.
  • Riesz fractional derivatives, defined by Lévy indices, are foundational.

Purpose of the Study:

  • To review one- and two-dimensional models for wave propagation in fractional media.
  • To outline soliton species generated by fractional models with nonlinear terms.
  • To summarize recent experimental findings in fractional optics.

Main Methods:

  • Review of existing literature on fractional wave propagation models.
  • Analysis of models based on Riesz fractional derivatives.
  • Examination of variational approximation for soliton analysis.

Main Results:

  • One- and two-dimensional models for linear and nonlinear wave propagation are presented.
  • Basic species of one-dimensional solitons in fractional media are outlined.
  • The variational approximation is shown to be effective for many cases.

Conclusions:

  • Fractional media models offer a framework for studying complex wave phenomena.
  • Soliton dynamics in fractional systems can be analyzed using established methods.
  • Experimental advancements validate theoretical models of fractional wave propagation.