Bracket structures for adjoint-symmetries and symmetries, and their applications

Abstract

Symmetries of a partial differential equation (PDE) can be defined 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 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 like a pull-back of the symmetry commutator bracket and has the properties of a Lie bracket. In the case of variational PDEs, adjoint-symmetries coincide with symmetries, and the linear actions themselves constitute new bilinear symmetry brackets which differ from the commutator bracket when acting on non-variational symmetries. Several examples of nonlinear PDEs are used to illustrate all of the results.

Geometrical formulation for adjoint-symmetries of partial differential equations

“…The second application is a geometrical derivation of three actions of symmetries on adjoint-symmetries. These symmetries actions have been obtained in recent work using an algebraic point of view [8]. They will be shown here to arise from Cartan's formula for the Lie derivative of an adjoint-symmetry 1-form (3.7).…”

“…For any PDE system (3.1), its set of adjoint-symmetries is a linear space, and as shown in Ref. [8], symmetries of the PDE system have three different actions on this space. The primary symmetry action can be derived from the Lie derivative of an adjointsymmetry 1-form with respect to a symmetry vector field.…”

“…It will provide a geometrical derivation of a well-known formula that generates a conservation law from a pair consisting of a symmetry and an adjoint-symmetry [3,1]. It also will yield three different actions of symmetries on adjoint-symmetries from Cartan's formula for the Lie derivative, providing a geometrical formulation of some recent work that used an algebraic viewpoint [8].…”

Roadmap of the Multiplier Method for Partial Differential Equations

“…Giving the relation between symmetries and adjoint-symmetries. The action of symmetries on adjoint-symmetries can also be derived in this geometrical formulation noting that for a given PDE system (78), the set of adjoint-symmetries is also linear space, and as shown in [57] symmetries of the PDE system have then three different types of actions in this space.…”