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Fractional White-Noise Limit and Paraxial Approximation for Waves in Random Media

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Abstract

This work is devoted to the asymptotic analysis of high frequency wave propagation in random media with long-range dependence. We are interested in two asymptotic regimes, that we investigate simultaneously: the paraxial approximation, where the wave is collimated and propagates along a privileged direction of propagation, and the white-noise limit, where random fluctuations in the background are well approximated in a statistical sense by a fractional white noise. The fractional nature of the fluctuations is reminiscent of the long-range correlations in the underlying random medium. A typical physical setting is laser beam propagation in turbulent atmosphere. Starting from the high frequency wave equation with fast non-Gaussian random oscillations in the velocity field, we derive the fractional Itô–Schrödinger equation, that is, a Schrödinger equation with potential equal to a fractional white noise. The proof involves a fine analysis of the backscattering and of the coupling between the propagating and evanescent modes. Because of the long-range dependence, classical diffusion-approximation theorems for equations with random coefficients do not apply, and we therefore use moment techniques to study the convergence.

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Authors and Affiliations

  1. Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France

    Christophe Gomez

  2. Department of Mathematics, Colorado State University, Fort Collins, CO, USA

    Olivier Pinaud

Authors
  1. Christophe Gomez
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  2. Olivier Pinaud
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Corresponding author

Correspondence to Christophe Gomez.

Additional information

Communicated by C. Le Bris

Olivier Pinaud acknowledges support from NSF CAREER Grant DMS-1452349.

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Gomez, C., Pinaud, O. Fractional White-Noise Limit and Paraxial Approximation for Waves in Random Media. Arch Rational Mech Anal 226, 1061–1138 (2017). https://doi.org/10.1007/s00205-017-1150-z

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  • DOI: https://doi.org/10.1007/s00205-017-1150-z

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