### Abstract

The phenomenon of self-induced transparency (SIT) is reinterpreted in the context of competition between randomness, nonlinearity and dispersion, and furthermore the problem is recast to show that it is isomorphic to a problem of the nonlinear Schroedinger (NLS) type with a random potential in which the randomness is manifested spatially. It is shown that, under mild assumptions, the SIT result continues to hold when we replace the uniform medium of inhomogeneously broadened two-level atoms by a series of intervals in each of which the frequency mismatch is randomly chosen from some distribution. The exact solution of this problem confirms and reveals the reason for the fact that nonlinearity can help improve the transparency of the medium. Also, the small-amplitude, almost monochromatic limit of SIT is taken and results in a complex envelope equation which turns out to be an exactly integrable combination of NLS and a modified SIT equation. Finally, some generalizations are made to describe a broad class of integrable systems which combine randomness, nonlinearity and dispersion.

Original language | English (US) |
---|---|

Pages (from-to) | 267-286 |

Number of pages | 20 |

Journal | IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications) |

Volume | 63 |

Issue number | 3 |

State | Published - Dec 1999 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Competition between nonlinearity, dispersion and randomness in signal propagation.** / Komarova, Natalia L.; Newell, Alan C.

Research output: Contribution to journal › Article

*IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)*, vol. 63, no. 3, pp. 267-286.

}

TY - JOUR

T1 - Competition between nonlinearity, dispersion and randomness in signal propagation

AU - Komarova, Natalia L.

AU - Newell, Alan C

PY - 1999/12

Y1 - 1999/12

N2 - The phenomenon of self-induced transparency (SIT) is reinterpreted in the context of competition between randomness, nonlinearity and dispersion, and furthermore the problem is recast to show that it is isomorphic to a problem of the nonlinear Schroedinger (NLS) type with a random potential in which the randomness is manifested spatially. It is shown that, under mild assumptions, the SIT result continues to hold when we replace the uniform medium of inhomogeneously broadened two-level atoms by a series of intervals in each of which the frequency mismatch is randomly chosen from some distribution. The exact solution of this problem confirms and reveals the reason for the fact that nonlinearity can help improve the transparency of the medium. Also, the small-amplitude, almost monochromatic limit of SIT is taken and results in a complex envelope equation which turns out to be an exactly integrable combination of NLS and a modified SIT equation. Finally, some generalizations are made to describe a broad class of integrable systems which combine randomness, nonlinearity and dispersion.

AB - The phenomenon of self-induced transparency (SIT) is reinterpreted in the context of competition between randomness, nonlinearity and dispersion, and furthermore the problem is recast to show that it is isomorphic to a problem of the nonlinear Schroedinger (NLS) type with a random potential in which the randomness is manifested spatially. It is shown that, under mild assumptions, the SIT result continues to hold when we replace the uniform medium of inhomogeneously broadened two-level atoms by a series of intervals in each of which the frequency mismatch is randomly chosen from some distribution. The exact solution of this problem confirms and reveals the reason for the fact that nonlinearity can help improve the transparency of the medium. Also, the small-amplitude, almost monochromatic limit of SIT is taken and results in a complex envelope equation which turns out to be an exactly integrable combination of NLS and a modified SIT equation. Finally, some generalizations are made to describe a broad class of integrable systems which combine randomness, nonlinearity and dispersion.

UR - http://www.scopus.com/inward/record.url?scp=0033330257&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033330257&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0033330257

VL - 63

SP - 267

EP - 286

JO - IMA Journal of Applied Mathematics

JF - IMA Journal of Applied Mathematics

SN - 0272-4960

IS - 3

ER -