A Mathematical Justification of the Isobe–Kakinuma Model for Water Waves with and without Bottom Topography

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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(δ4 N + 2) in the case of the flat bottom and of order O(δ4 [ N / 2 ] + 2) in the case of variable bottoms.

Original languageEnglish
Pages (from-to)1985-2018
Number of pages34
JournalJournal of Mathematical Fluid Mechanics
Volume20
Issue number4
DOIs
Publication statusPublished - 2018 Dec 1

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water waves
Water waves
Water Waves
Topography
Justification
topography
Model
Euler-Lagrange Equations
Shallow Water
First order differential equation
shallow water
Small Parameter
partial differential equations
Justify
Partial differential equations
integers
Error Estimates
Partial differential equation
Non-negative
Wavelength

ASJC Scopus subject areas

  • Mathematical Physics
  • Condensed Matter Physics
  • Computational Mathematics
  • Applied Mathematics

Cite this

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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(δ4 N + 2) in the case of the flat bottom and of order O(δ4 [ N / 2 ] + 2) in the case of variable bottoms.",
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