### Abstract

Inverse spectral theory is used to prescribe and study equations for the slow modulations of N‐phase wave trains for the Korteweg‐de Vries (KdV) equation. An invariant representation of the modulational equations is deduced. This representation depends upon certain differentials on a Riemann surface. When evaluated near ∞ on the surface, the invariant representation reduces to averaged conservations laws; when evaluated near the branch points, the representation shows that the simple eigenvalues provide Riemann invariants for the modulational equations. Integrals of the invariant representation over certain cycles on the Riemann surface yield “conservation of waves.” Explicit formulas for the characteristic speeds of the modulational equations are derived. These results generalize known results for a single‐phase traveling wave, and indicate that complete integrability can induce enough structure into the modulational equations to diagonalize (in the sense of Riemann invariants) their first‐order terms.

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

Pages (from-to) | 739-784 |

Number of pages | 46 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 33 |

Issue number | 6 |

DOIs | |

State | Published - 1980 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*33*(6), 739-784. https://doi.org/10.1002/cpa.3160330605

**Multiphase averaging and the inverse spectral solution of the Korteweg—de Vries equation.** / Flaschka, Hermann; Forest, M. G.; McLaughlin, D. W.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 33, no. 6, pp. 739-784. https://doi.org/10.1002/cpa.3160330605

}

TY - JOUR

T1 - Multiphase averaging and the inverse spectral solution of the Korteweg—de Vries equation

AU - Flaschka, Hermann

AU - Forest, M. G.

AU - McLaughlin, D. W.

PY - 1980

Y1 - 1980

N2 - Inverse spectral theory is used to prescribe and study equations for the slow modulations of N‐phase wave trains for the Korteweg‐de Vries (KdV) equation. An invariant representation of the modulational equations is deduced. This representation depends upon certain differentials on a Riemann surface. When evaluated near ∞ on the surface, the invariant representation reduces to averaged conservations laws; when evaluated near the branch points, the representation shows that the simple eigenvalues provide Riemann invariants for the modulational equations. Integrals of the invariant representation over certain cycles on the Riemann surface yield “conservation of waves.” Explicit formulas for the characteristic speeds of the modulational equations are derived. These results generalize known results for a single‐phase traveling wave, and indicate that complete integrability can induce enough structure into the modulational equations to diagonalize (in the sense of Riemann invariants) their first‐order terms.

AB - Inverse spectral theory is used to prescribe and study equations for the slow modulations of N‐phase wave trains for the Korteweg‐de Vries (KdV) equation. An invariant representation of the modulational equations is deduced. This representation depends upon certain differentials on a Riemann surface. When evaluated near ∞ on the surface, the invariant representation reduces to averaged conservations laws; when evaluated near the branch points, the representation shows that the simple eigenvalues provide Riemann invariants for the modulational equations. Integrals of the invariant representation over certain cycles on the Riemann surface yield “conservation of waves.” Explicit formulas for the characteristic speeds of the modulational equations are derived. These results generalize known results for a single‐phase traveling wave, and indicate that complete integrability can induce enough structure into the modulational equations to diagonalize (in the sense of Riemann invariants) their first‐order terms.

UR - http://www.scopus.com/inward/record.url?scp=84980178052&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84980178052&partnerID=8YFLogxK

U2 - 10.1002/cpa.3160330605

DO - 10.1002/cpa.3160330605

M3 - Article

VL - 33

SP - 739

EP - 784

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 6

ER -