### Abstract

The classical shallow water equations express the change with time of the height h and the velocity ν of a 1-dimensional fluid: νξ νt+ νξ νx+ νh νx=0. νh νx+ νhν νx=0. They possess an infinite number of integrals of motion due to Benney [1973] and can be written in Hamiltonian form relative to a symplectic structure introduced by Manin [1978]. The present paper deals with their complete integrability up to the advent of shocks. This is proved in the small under an extra assumption satisfied by most height-velocity pairs: that hh′ = ± ν′ only at isolated points.

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

Pages (from-to) | 253-260 |

Number of pages | 8 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 4 |

Issue number | 2 |

DOIs | |

State | Published - 1982 |

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### ASJC Scopus subject areas

- Applied Mathematics
- Statistical and Nonlinear Physics

### Cite this

*Physica D: Nonlinear Phenomena*,

*4*(2), 253-260. https://doi.org/10.1016/0167-2789(82)90066-5

**The classical shallow water equations : Symplectic geometry.** / Cavalcante, J.; McKean, H. P.

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol. 4, no. 2, pp. 253-260. https://doi.org/10.1016/0167-2789(82)90066-5

}

TY - JOUR

T1 - The classical shallow water equations

T2 - Symplectic geometry

AU - Cavalcante, J.

AU - McKean, H. P.

PY - 1982

Y1 - 1982

N2 - The classical shallow water equations express the change with time of the height h and the velocity ν of a 1-dimensional fluid: νξ νt+ νξ νx+ νh νx=0. νh νx+ νhν νx=0. They possess an infinite number of integrals of motion due to Benney [1973] and can be written in Hamiltonian form relative to a symplectic structure introduced by Manin [1978]. The present paper deals with their complete integrability up to the advent of shocks. This is proved in the small under an extra assumption satisfied by most height-velocity pairs: that hh′ = ± ν′ only at isolated points.

AB - The classical shallow water equations express the change with time of the height h and the velocity ν of a 1-dimensional fluid: νξ νt+ νξ νx+ νh νx=0. νh νx+ νhν νx=0. They possess an infinite number of integrals of motion due to Benney [1973] and can be written in Hamiltonian form relative to a symplectic structure introduced by Manin [1978]. The present paper deals with their complete integrability up to the advent of shocks. This is proved in the small under an extra assumption satisfied by most height-velocity pairs: that hh′ = ± ν′ only at isolated points.

UR - http://www.scopus.com/inward/record.url?scp=0000017404&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000017404&partnerID=8YFLogxK

U2 - 10.1016/0167-2789(82)90066-5

DO - 10.1016/0167-2789(82)90066-5

M3 - Article

VL - 4

SP - 253

EP - 260

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 2

ER -