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

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.

Burgers turbulence in the Fermi-Pasta-Ulam-Tsingou chain

“…-Model, initial conditions and continuum approximation. All the details of the following analytical derivation are reported in the Supplemental Material [29] (see [32] for the mathematical framework).…”

“…Such a "gradient catastrophe" implies a transfer of energy to the highest Fourier modes of wavelength ∼ 1/N , so that a global continuum limit no longer holds after the shock. For a correct continuum description of the shock region, higher order derivatives of the fields must be taken into account, which replaces the Burgers equations with a pair of KdV equations [32,[36][37][38]. However, far from the shock region, the Burgers equation still describes the FPUT dynamics.…”