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(δ 2 ), 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(δ 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(δ 6 ).and the action function is

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

“…in the case (H1), δ 4[N/2]+2 in the case (H2), as was expected by the analysis of linear dispersion relations, where η IK is the solution of the Isobe-Kakinuma model. As mentioned before, in the case N = 1 and p 1 = 2 with flat bottom, this error estimate was shown by T. Iguchi [7].…”

“…In this paper, we will show that this is correct even for the nonlinear problem with variable bottoms. In a particular case, that is, in the case N = 1 and p 1 = 2 with the flat bottom, this was shown by T. Iguchi [7]. Therefore, this paper is a generalization of his results.…”

Solvability of the Initial Value Problem to the Isobe–Kakinuma Model for Water Waves

“…This relation (1.5) implies that the Isobe-Kakinuma model gives a good approximation of the basic equations for water waves in the shallow water regime h|ξ| ≪ 1. In fact, T. Iguchi [7] gave a mathematically rigorous justification of the Isobe-Kakinuma model in the case of the flat bottom with the choice N = 1 and p 1 = 2. He showed that the Isobe-Kakinuma model gives a higher order shallow water approximation for water waves with an error of order O(δ 6 ), where δ is a small nondimensional parameter defined as the ratio of the mean depth h to the typical wave length.…”

A Hamiltonian Structure of the Isobe–Kakinuma Model for Water Waves

“…As aforementioned, it was shown in [11,12] that the Isobe-Kakinuma model (1.9) is a higher order shallow water approximation for the water wave problem in the strongly nonlinear regime.…”