## Abstract

Problems that arise when the perturbative analysis of the Boussinesq equations for the flow in a shallow-water layer over a flat horizontal bottom is extended beyond the KdV approximation are studied. It is shown that the solution unavoidably contains a third-order non-local term. In the single-soliton case, this term is localized along the soliton trajectory. This ensures that the unidirectional solution is mass conserving at least through third order. In the multiple-soliton case, the non-local term represents inelastic soliton interactions; it does not vanish along a strip in the x - t plane, which emanates form the soliton-collision region. Due to the freedom inherent in the perturbation analysis, this term may be incorporated either in the solution for the surface elevation, or in the solution for the horizontal velocity component. With the first choice, through third order, the unidirectional solution is mass conserving, whereas with the second-it is not.

Original language | English |
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Pages (from-to) | 147-161 |

Number of pages | 15 |

Journal | Advances and Applications in Fluid Mechanics |

Volume | 11 |

Issue number | 2 |

State | Published - 1 Apr 2012 |

## Keywords

- Boussinesq approximation
- Mass conservation
- Shallow-water equations
- Soliton interactions

## ASJC Scopus subject areas

- Mechanical Engineering