WIAS Preprint No. 2512, (2018)

Generalized Lighthill criterion for the modulation instability


  • Amiranashvili, Shalva
    ORCID: 0000-0002-8132-882X
  • Tobisch, Elena

2010 Mathematics Subject Classification

  • 78A40 78A60

2008 Physics and Astronomy Classification Scheme

  • 42.25.Bs 42.65.Sf 42.65.Wi 42.81.Dp


  • Lighthill criterion, modilational instability, generalized nonlinear Schrödinger equation, four wave mixing




An universal modulation instability is subject to Lighthill criterion: nonlinearity and dispersion should make opposite contributions to the wave frequency. Recent studies of wave instabilities in optical fibers with the minimum chromatic dispersion revealed situations in which the criterion is violated and fast unstable modulations appear due to the four wave mixing process. We derive a generalized criterion, it applies to an arbitrary dispersion and to both slow and fast unstable modulations. Since the fast modulations depend on nonlinear dispersion, we also demonstrate how to describe them in the framework of a single generalized nonlinear Schrödinger equation.

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