Isobe-Kakinuma model for water waves as a higher order shallow water approximation

Abstract

We justify rigorously an Isobe-Kakinuma model for water waves as a higher order shallow water approximation in the case of a flat bottom. It is known that the full water wave equations are approximated by the shallow water equations with an error of order $O({\delta }^2)$, where $\delta$ is a small nondimensional parameter defined as the ratio of the mean depth to the typical wavelength. The Green-Naghdi equations are known as higher order approximate equations to the water wave equations with an error of order $O({\delta }^4)$. In this paper we show that the Isobe-Kakinuma model is a much higher order approximation to the water wave equations with an error of order $O({\delta }^6)$.

25 pages

References in corpus (2)

• Hamiltonian formulation of the extended Green-Naghdi equations
• Hamiltonian structure for two-dimensional extended Green-Naghdi equations

Cited by in corpus (7)

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• A Hamiltonian structure of the Isobe-Kakinuma model for water waves
• Solitary wave solutions to the Isobe-Kakinuma model for water waves
• A mathematical analysis of the Kakinuma model for interfacial gravity waves. Part II: Justification as a shallow water approximation
• A mathematical analysis of the Kakinuma model for interfacial gravity waves. Part I: Structures and well-posedness