Symmetry actions and brackets for adjoint-symmetries. I: Main results and applications

TL;DR

This paper explores the algebraic structure of adjoint-symmetries in PDEs, introducing symmetry actions and brackets that generalize Lie algebra concepts without requiring variational structures, with applications to integrable systems.

Contribution

It introduces new symmetry actions and bilinear brackets for adjoint-symmetries applicable to general PDEs, extending the algebraic framework beyond variational systems.

Findings

Constructed bilinear adjoint-symmetry brackets including a Lie bracket

Identified symmetry actions encoding pre-sympletic and symplectic structures

Applied results to the potential form of the generalized KdV equation

Abstract

Infinitesimal symmetries of a partial differential equation (PDE) can be defined algebraically as the solutions of the linearization (Frechet derivative) equation holding on the space of solutions to the PDE, and they are well-known to comprise a linear space having the structure of a Lie algebra.Solutions of the adjoint linearization equation holding on the space of solutions to the PDE are called adjoint-symmetries. Their algebraic structure for general PDE systems is studied herein. This is motivated by the correspondence between variational symmetries and conservation laws arising from Noether's theorem, which has a well-known generalization to non-variational PDEs, where infinitesimal symmetries are replaced by adjoint-symmetries, and variational symmetries are replaced by multipliers (adjoint-symmetries satisfying a certain Euler-Lagrange condition). Several main results are obtained. Symmetries are shown to have three different linear actions on the linear space of adjoint-symmetries. These linear actions are used to construct bilinear adjoint-symmetry brackets, one of which is a pull-back of the symmetry commutator bracket and has the properties of a Lie bracket. The brackets do not use or require the existence of any local variational structure (Hamiltonian or Lagrangian) and thus apply to general PDE systems. One of the symmetry actions is shown to encode a pre-sympletic (Noether) operator and a symplectic 2-form, which lead to the construction of a symplectic 2-form and Poisson bracket for evolution systems. The generalized KdV equation in potential form is used to illustrate all of the results.