A formula for symmetry recursion operators from non-variational symmetries of partial differential equations

Anco, Stephen C.; Wang, Bao

doi:10.1007/s11005-021-01413-1

Cited by 10 publications

(31 citation statements)

References 15 publications

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“…It will also be interesting to develop fully the use of adjoint-symmetries in the study of specific PDE systems, as outlined in the introduction: finding exact solutions, detecting and finding mappings into a target class of PDEs, and detecting integrability, which are counterparts of some important uses of symmetries. Another use of adjoint-symmetries, which has been introduced very recently [7], is for finding pre-symplectic operators.…”

Section: Discussionmentioning

confidence: 99%

Geometrical formulation for adjoint-symmetries of partial differential equations

Anco¹,

Bao

2020

Preprint

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A geometrical formulation for adjoint-symmetries as 1-forms is studied for general partial differential equations (PDEs), which provides a dual counterpart of the geometrical meaning of symmetries as tangent vector fields on the solution space of a PDE. Two applications of this formulation are presented. Additionally, for systems of evolution equations, adjoint-symmetries are shown to have another geometrical formulation given by 1-forms that are invariant under the flow generated by the system on the solution space. This result is generalized to systems of evolution equations with spatial constraints, where adjoint-symmetry 1-forms are shown to be invariant up to a functional multiplier of a normal 1-form associated to the constraint equations. All of the results are applicable to the PDE systems of interest in applied mathematics and mathematical physics.

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Section: Discussionmentioning

confidence: 99%

Geometrical formulation for adjoint-symmetries of partial differential equations

Anco¹,

Bao

2020

Preprint

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“…The preceding results are a full and complete generalisation of the symmetry actions derived for scalar PDEs in [11]. They will now be summarised, and then, some of their consequences will be developed.…”

Section: Action Of Symmetries On Adjoint-symmetriesmentioning

confidence: 91%

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

Anco

2022

Eur. J. Appl. Math

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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.

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“…This adjoint-symmetry comes from the scaling symmetry of the KdV equation in potential form, as shown in Ref. [11]. In particular, each symmetry of that equation is an adjoint-symmetry of the KdV equation.…”

Section: Examplesmentioning

confidence: 95%

“…from Symm G into AdjSymm G , which constitutes a generalized pre-symplectic operator [11] in analogy with symplectic operators that map symmetries into adjoint-symmetries for Hamiltonian systems. For a fixed adjoint-symmetry Q A , S Q will have an inverse S −1 Q which is defined modulo its kernel, ker(S Q ) ⊂ Symm G , and which acts on the linear subspace given by its range,…”

Section: Bracket Structures For Adjoint-symmetriesmentioning

confidence: 99%

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

Anco¹

2020

Preprint

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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.

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A formula for symmetry recursion operators from non-variational symmetries of partial differential equations

Cited by 10 publications

References 15 publications

Geometrical formulation for adjoint-symmetries of partial differential equations

Geometrical formulation for adjoint-symmetries of partial differential equations

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

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

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