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

Abstract

Infinitesimal symmetries of a partial differential equation (PDE) can be defined algebraically as the solutions of the linearisation (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 linearisation 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 modern generalisation 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-symplectic (Noether) operator, which leads to the construction of symplectic 2-form and Poisson bracket for evolution systems. The generalised KdV equation in potential form is used to illustrate all of the results.

Symmetry actions and brackets for adjoint-symmetries. II: Physical examples

“…This 2-form is symplectic, namely dω Q = 0, as proven in [3]. The formal inverse of the Noether operator (2.27) defines a pre-Hamiltonian (inverse Noether) operator J −1 which maps adjoint-symmetries into symmetries.…”

“…, where δ/δu denotes the variational derivative, namely δF/δu α = E u α (f ). In particular, the Jacobi identity for this bracket holds as a consequence of closure of the symplectic 2-form (see [3], and also [23] for a related general result).…”

“…The mathematical setting will be calculus in jet space [23], which is summarised in the appendix of [3]. Partial derivatives and total derivatives are denoted using a coordinate notation.…”

“…As explained in [3], some technical conditions related to local solvability, involutivity, and existence of a solved form for leading derivatives will be assumed on PDE systems, which are called regular systems [2]. These conditions hold for essentially all PDE systems of interest in physical applications.…”

“…Fourth, a Noether (pre-symplectic) operator will be shown to arise directly from the symmetry actions in examples (3) to (5), using an adjoint-symmetry that is not a multiplier. In examples (3) and (4), this operator yields a symplectic 2-form and a corresponding Hamiltonian structure. In example (5), the Noether operator yields a Lagrangian structure.…”