A Hamiltonian structure of the Isobe-Kakinuma model for water waves

TL;DR

This paper demonstrates that the Isobe-Kakinuma model for water waves possesses a Hamiltonian structure similar to the full water wave problem, with its Hamiltonian being a higher order shallow water approximation.

Contribution

The paper reveals the Hamiltonian structure of the Isobe-Kakinuma model and connects it to the full water wave problem through a higher order shallow water approximation.

Findings

The Isobe-Kakinuma model has a Hamiltonian structure analogous to Zakharov's formulation.

The Hamiltonian of the model approximates the full water wave Hamiltonian at higher order.

The model provides a new perspective on water wave dynamics with Hamiltonian formalism.

Abstract

We consider the Isobe-Kakinuma model for water waves, which is obtained as the system of Euler-Lagrange equations for a Lagrangian approximating Luke's Lagrangian for water waves. We show that the Isobe-Kakinuma model also enjoys a Hamiltonian structure analogous to the one exhibited by V. E. Zakharov on the full water wave problem and, moreover, that the Hamiltonian of the Isobe-Kakinuma model is a higher order shallow water approximation to the one of the full water wave problem.