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

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.