Isobe–Kakinuma Model for Water Waves

Abstract

We consider the initial value problem to the Isobe–Kakinuma model for water waves. The Isobe–Kakinuma model is the Euler–Lagrange equation for an approximate Lagrangian which is derived from Luke’s Lagrangian for water waves by approximating the velocity potential in the Lagrangian. The Isobe–Kakinuma model consists of \((N+1)\) second-order and a first-order partial differential equations, where N is a nonnegative integer, and is classified into a system of nonlinear dispersive equations. Since the hypersurface \(t = 0\) is characteristic for the Isobe–Kakinuma model, the initial data have to be restricted in an infinite-dimensional manifold for the existence of the solution. Under this necessary condition and a sign condition, the initial value problem turns out to be well-posed locally in time in Sobolev spaces. Then, we present a rigorous justification of the Isobe–Kakinuma model as a higher order shallow water approximation in the strongly nonlinear regime. The error between the solutions of the Isobe–Kakinuma model and of the full water wave problem turns out to be of order \(O(\delta ^{4N+2})\) in the case of the flat bottom and of order \(O(\delta ^{4[N/2]+2})\) in the case of a variable bottom, where \(\delta \) is a nondimensional parameter given by the ratio of the mean depth to the typical wavelength and represents shallowness of the water. Therefore, the Isobe–Kakinuma model is a much higher approximation than the well-known Green–Naghdi equations.