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We justify rigorously an Isobe-Kakinuma model for water waves as a higherorder shallow water approximation in the case of a flat bottom. It is knownthat the full water wave equations are approximated by the shallow waterequations with an error of order $O(\delta^2)$, where $\delta$ is a smallnondimensional parameter defined as the ratio of the mean depth to the typicalwavelength. The Green-Naghdi equations are known as higher order approximateequations to the water wave equations with an error of order $O(\delta^4)$. Inthis paper we show that the Isobe-Kakinuma model is a much higher orderapproximation to the water wave equations with an error of order $O(\delta^6)$.

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