### Abstract

We consider the Isobe–Kakinuma model for water waves in both cases of the flat and the variable bottoms. The Isobe–Kakinuma model is a system of Euler–Lagrange equations for an approximate Lagrangian which is derived from Luke’s Lagrangian for water waves by approximating the velocity potential in the Lagrangian appropriately. The Isobe–Kakinuma model consists of (N+ 1) second order and a first order partial differential equations, where N is a nonnegative integer. We justify rigorously the Isobe–Kakinuma model as a higher order shallow water approximation in the strongly nonlinear regime by giving an error estimate between the solutions of the Isobe–Kakinuma model and of the full water wave problem in terms of the small nondimensional parameter δ, which is the ratio of the mean depth to the typical wavelength. It turns out that the error is of order O(δ^{4}
^{N}
^{+}
^{2}) in the case of the flat bottom and of order O(δ^{4}
^{[}
^{N}
^{/}
^{2}
^{]}
^{+}
^{2}) in the case of variable bottoms.

Original language | English |
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Pages (from-to) | 1985-2018 |

Number of pages | 34 |

Journal | Journal of Mathematical Fluid Mechanics |

Volume | 20 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2018 Dec 1 |

### ASJC Scopus subject areas

- Mathematical Physics
- Condensed Matter Physics
- Computational Mathematics
- Applied Mathematics