• DOI:10.1137/060668882
• Corpus ID: 12034674

Self-Averaging from Lateral Diversity in the Itô-Schrödinger Equation

@article{Papanicolaou2007SelfAveragingFL,
  title={Self-Averaging from Lateral Diversity in the It{\^o}-Schr{\"o}dinger Equation},
  author={George C Papanicolaou and Lenya Ryzhik and Knut S{\o}lna},
  journal={Multiscale Model. Simul.},
  year={2007},
  volume={6},
  pages={468-492}
}

• G. Papanicolaou, L. Ryzhik, K. Sølna
• Published 7 June 2007
• Physics, Mathematics
• Multiscale Model. Simul.

We consider the random Schrodinger equation as it arises in the paraxial regime for wave propagation in random media. In the white noise limit it becomes the Ito–Schrodinger stochastic partial differential equation which we analyze here in the high frequency regime. We also consider the large lateral diversity limit where the typical width of the propagating beam is large compared to the correlation length of the random medium. We use the Wigner transform of the wave field and show that it becomes deterministic in the large diversity limit when integrated against test functions. This is the self-averaging property of the Wigner transform. It follows easily when the support of the test functions is of the order of the beam width. We also show with a more detailed analysis that the limit is deterministic when the support of the test functions tends to zero but is large compared to the correlation length.