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Singular and regular gap solitons between three dispersion curves.

Roger Grimshaw1, Boris A Malomed, Georg A Gottwald

  • 1Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, United Kingdom. R.H.J.Grimshaw@lboro.ac.uk

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 22, 2002
PubMed
Summary

This study introduces a general wave model with three coupled waves, revealing two distinct soliton families: regular solitons and singular cuspons or peakons. Numerical analysis confirms the stability of these novel soliton types.

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Area of Science:

  • Nonlinear physics
  • Wave phenomena
  • Soliton theory

Background:

  • Wave-envelope systems with coupled modes are crucial in various physical domains.
  • Linear dispersion relations with nearly coincident intersection points present unique challenges.
  • Existing models often focus on two-wave interactions, leaving three-wave systems less explored.

Purpose of the Study:

  • To develop a general model for wave-envelope systems with three coupled branches and two spectral gaps.
  • To investigate the nonlinear behavior and soliton solutions in this three-wave system.
  • To explore the existence and stability of different soliton types, including regular solitons, cuspons, and peakons.

Main Methods:

  • Formulation of a general wave-envelope model incorporating strong linear coupling between two waves and a third coupled wave.

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  • Nonlinear analysis focusing on zero-velocity solitons.
  • Derivation of analytical solutions, with and without self-phase modulation (SPM) terms.
  • Numerical simulations to assess the stability of the identified soliton solutions.

    • Main Results:

    • Discovery of two distinct soliton families: regular solitons and singular cuspons (without SPM) or peakons (in the limit of vanishing coupling or with SPM).
    • Analytical solutions obtained for both regular solitons and cuspons/peakons.
    • Numerical stability tests indicate that regular solitons, cuspons, and peakons can all be stable.
    • Simulations confirm the coexistence and stability of regular solitons and peakons when SPM terms are included.

    Conclusions:

    • The developed model successfully describes complex wave interactions in systems with three coupled branches.
    • The existence of multiple, distinct soliton families (regular, cuspons, peakons) highlights novel nonlinear dynamics.
    • These findings have potential applications in optical systems (waveguides, photonic-crystal fibers) and fluid dynamics (internal waves).