Publication: Variational operators, symplectic operators, and the cohomology of scalar evolution equations
No Thumbnail Available
Date
2019-06-04
Authors
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Springernature
Abstract
For a scalar evolution equation u(t) = K(t, x, u, u(x), . . . , u(2m+1)) with m >= 1, the cohomology space H-1,H-2() is shown to be isomorphic to the space of variational operators and an explicit isomorphism is given. The space of symplectic operators for u(t) = K for which the equation is Hamiltonian is also shown to be isomorphic to the space H-1,H-2() and subsequently can be naturally identified with the space of variational operators. Third order scalar evolution equations admitting a first order symplectic (or variational) operator are characterized. The variational operator (or symplectic) nature of the potential form of a bi-Hamiltonian evolution equation is also presented in order to generate examples of interest.
Description
Keywords
Inverse problem, Calculus, Bicomplexes, Variational bicomplex, Cohomology, Scalar evolution equation, Symplectic operator, Hamiltonian evolution equation, Mathematics, Physics
Citation
Collections
Metrikler