Hamiltonian field theory close to the wave equation: from Fermi-Pasta-Ulam to water waves

TL;DR

This paper explores the Hamiltonian field theory near the wave equation, revealing its equivalence to the Korteweg-de Vries hierarchy under polynomial perturbations, and clarifies the link between water waves and the Fermi-Pasta-Ulam system.

Contribution

It demonstrates the Hamiltonian structure's reduction to the KdV hierarchy near the wave equation and elucidates the connection between water wave theory and Fermi-Pasta-Ulam system.

Findings

Hamiltonian field theory near wave equation is equivalent to KdV hierarchy.

Polynomial perturbations preserve the Hamiltonian structure.

Connection established between water waves and Fermi-Pasta-Ulam system.

Abstract

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.