### Abstract

Detailed study is made of the way in which weak nonlinearities affect the statistical properties of a system of dispersive waves. Given that at some initial instant the spectral cumulants are sufficiently smooth it is shown that they will remain smooth to a zeroth order, save in one dimension where a discrete spectrum may eventually be generated. Of prime interest is the fact that on considering the long time behavior of the system, one is led to a sequence of closures for the zeroth order spectral functions. Apparent difficulties associated with the irretraceability of the solution are discussed. The structure of the closure equations depends on the asymptotic behavior of a class of singular integrals.

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

Pages (from-to) | 29-53 |

Number of pages | 25 |

Journal | Studies in Applied Mathematics |

Volume | 48 |

Issue number | 1 |

State | Published - Mar 1969 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Studies in Applied Mathematics*,

*48*(1), 29-53.

**RANDOM WAVE CLOSURES.** / BENNEY DJ, DJ; Newell, Alan C.

Research output: Contribution to journal › Article

*Studies in Applied Mathematics*, vol. 48, no. 1, pp. 29-53.

}

TY - JOUR

T1 - RANDOM WAVE CLOSURES

AU - BENNEY DJ, DJ

AU - Newell, Alan C

PY - 1969/3

Y1 - 1969/3

N2 - Detailed study is made of the way in which weak nonlinearities affect the statistical properties of a system of dispersive waves. Given that at some initial instant the spectral cumulants are sufficiently smooth it is shown that they will remain smooth to a zeroth order, save in one dimension where a discrete spectrum may eventually be generated. Of prime interest is the fact that on considering the long time behavior of the system, one is led to a sequence of closures for the zeroth order spectral functions. Apparent difficulties associated with the irretraceability of the solution are discussed. The structure of the closure equations depends on the asymptotic behavior of a class of singular integrals.

AB - Detailed study is made of the way in which weak nonlinearities affect the statistical properties of a system of dispersive waves. Given that at some initial instant the spectral cumulants are sufficiently smooth it is shown that they will remain smooth to a zeroth order, save in one dimension where a discrete spectrum may eventually be generated. Of prime interest is the fact that on considering the long time behavior of the system, one is led to a sequence of closures for the zeroth order spectral functions. Apparent difficulties associated with the irretraceability of the solution are discussed. The structure of the closure equations depends on the asymptotic behavior of a class of singular integrals.

UR - http://www.scopus.com/inward/record.url?scp=0014476464&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0014476464&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0014476464

VL - 48

SP - 29

EP - 53

JO - Studies in Applied Mathematics

JF - Studies in Applied Mathematics

SN - 0022-2526

IS - 1

ER -