Hamiltonian Field Theory Close to the Wave Equation: From Fermi-Pasta-Ulam to Water Waves

In the present work we analyse the structure of the Hamiltonian field theory in the neighbourhood of the wave equation q_{tt}= q_{xx} . We show that, restricting to “graded” polynomial perturbations in q_x , p and their space derivatives of higher order, the local field theory is equivalent, in the sense of the Hamiltonian normal form, to that of the Korteweg-de Vries hierarchy of second order. Within this framework, we explain the connection between the theory of water waves and the Fermi-Pasta-Ulam system.