Canonically conjugate variables for the periodic Camassa-Holm equation

TL;DR

This paper constructs conjugate variables for the periodic Camassa-Holm equation using spectral theory, enhancing the understanding of its Hamiltonian structure.

Contribution

It introduces a novel set of conjugate variables for both Poisson brackets of the Camassa-Holm equation based on spectral theory.

Findings

Conjugate variables are explicitly constructed for the equation.

The approach clarifies the Hamiltonian structure of the system.

Spectral theory is effectively applied to identify these variables.

Abstract

The Camassa-Holm shallow water equation is known to be Hamiltonian with respect to two compatible Poisson brackets. A set of conjugate variables is constructed for both brackets using spectral theory.