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

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(δ²), where δ 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(δ⁴). 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(δ⁶).