Variational Operators, Symplectic Operators, and the Cohomology of Scalar Evolution Equations

Fels M. E. , Yasar E.

JOURNAL OF NONLINEAR MATHEMATICAL PHYSICS, vol.26, no.4, pp.604-649, 2019 (Peer-Reviewed Journal) identifier identifier

  • Publication Type: Article / Article
  • Volume: 26 Issue: 4
  • Publication Date: 2019
  • Doi Number: 10.1080/14029251.2019.1640470
  • Journal Indexes: Science Citation Index Expanded, Scopus
  • Page Numbers: pp.604-649


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.