Science and Mathematics Faculty Publications

An Integrable Model of Stable:Unstable Wave Coupling Phenomena

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Physica D





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We report instability structures and nonlinear phenomena that arise when unstable and stable nonlinear wave fields are coupled nonlinearly. This interaction is modelled with an integrable system of cubic nonlinear Schrödinger (NLS) equations and plane wave data. The linearized analysis is straightforward, and robust to non-integrable perturbations. The coupled nonlinear Schrödinger (CNLS) model is chosen for several purposes: to contrast previous results on wave coupling phenomena for two stable or two unstable NLS fields; to exploit the rare tools of integrability to explicitly construct and visualize nonlinear homoclinic manifolds associated with these instabilities; and to work out results for two coupled scalar fields as a model for potential applications of these observations to wave-division multiplexing of optical pulse propagation. From earlier work of Forest, McLaughlin, Muraki and Wright (2000), a nonlinear coupling instability arises between two stable or two unstable scalar NLS fields, localized in wavenumbers which scale proportional to the background wave. Here we show the coupling of a stable and an unstable NLS field creates instability structures at wavelengths inversely proportional to the background. Two dramatic phenomena result for sufficiently longwave background data:

  • a combined energy transfer and pulse compression mechanism: a focusing plane wave, coupled to a weaker defocusing field, transfers energy at an exponential rate into the defocusing field at short wavelengths;

  • a mechanism for stabilization of the modulational instability of focusing NLS fields: longwave instabilities of a focusing plane wave can be stabilized by coupling to a stronger defocusing plane wave.


35Q55, 58F07, 02.30.Jr, 42.65.Sf, Plane waves, Wave coupling, Instabilities structures, Cubic Schrödinger equation