Invariant Reduction for Partial Differential Equations. III: Poisson Brackets

TL;DR

This paper demonstrates how invariant reduction of PDEs leads to finite-dimensional Hamiltonian systems inheriting Poisson structures, preserving integrability and constants of motion.

Contribution

It introduces a general mechanism for invariant reduction that consistently transfers Hamiltonian structures to reduced systems across various symmetry types.

Findings

Reduced systems inherit local Hamiltonian operators

Inherited Poisson brackets match original brackets up to sign

Examples show inherited structures imply integrability

Abstract

We show that, under suitable conditions, finite-dimensional systems describing invariant solutions of partial differential equations (PDEs) inherit local Hamiltonian operators through the mechanism of invariant reduction, which applies uniformly to point, contact, and higher symmetries. The inherited operators endow the reduced systems with Poisson bivectors that relate constants of motion to symmetries. Applying the same mechanism to invariant conservation laws, we further show that the induced Poisson brackets agree with those of the original systems, up to sign. The results are illustrated by two examples in which the inherited Poisson brackets and inherited constants of motion yield integrability of the reduced systems. The construction is independent of the choice of an $\ell$-normal inclusion of a PDE system into jet spaces.