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

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(δ 4N +2 ) in the case of the flat bottom and of order O(δ 4[N/2]+2 ) in the case of variable bottoms.

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.…”

“…Theorem 2.4 in [12] in fact states the stronger result that the difference between exact solutions of the water wave problem obtained in [10,18] and the corresponding solutions of the Isobe-Kakinuma model is bounded with the same order of precision as above on the relevant timescale.…”

“…Y. Murakami and T. Iguchi [22] and R. Nemoto and T. Iguchi [23] showed that the initial value problem to the Isobe-Kakinuma model (1.9) is well-posed locally in time in a class of initial data for which the relations (2.1) and a generalized Rayleigh-Taylor sign condition are satisfied. Moreover, T. Iguchi [11,12] showed that the Isobe-Kakinuma model (1.9) is a higher order shallow water approximation for the water wave problem in the strongly nonlinear regime. The Isobe-Kakinuma model (1.9) has also a conserved energy, which is the total energy given by…”

Solitary wave solutions to the Isobe-Kakinuma model for water waves

“…For the derivation and basic properties of this model, we refer to Y. Murakami and T. Iguchi [14] and R. Nemoto and T. Iguchi [15]. Moreover, it was shown by T. Iguchi [5,6] that the Isobe-Kakinuma model (1.3) is a higher order shallow water approximation for the water wave problem in the strongly nonlinear regime. We note also that the Isobe-Kakinuma model (1.3) in the case N = 0 is exactly the same as the shallow water equations.…”

“…We note that even in this simplest case the Isobe-Kakinuma model gives a better approximation than the well-known Green-Naghdi equations in the shallow water and strongly nonlinear regime. See T. Iguchi [5,6]. Numerical analysis suggests that there exists a critical value of δ given approximately by (1.8) δ c = 0.62633493 such that for any δ ∈ (0, δ c ), the Isobe-Kakinuma model (7.1) admits a smooth solitary wave solution and that this family of waves converges to a solitary one of extreme form with a shape crest as δ ↑ δ c .…”