Variational Operators, Symplectic Operators, and the Cohomology of Scalar Evolution Equations. Issue 4 (2nd October 2019)

• Record Type: Journal Article
• Authors: Fels, M.E.; Yaşar, E.
• Is Part Of: Journal of nonlinear mathematical physics. Volume 26:Issue 4(2019)
• Page Start: 604
• Page End: 649
• Publication Date: 2019-10-02
• Subjects: Variational Bicomplex -- Cohomology -- Scalar Evolution Equation -- Symplectic Operator -- Hamiltonian Evolution Equation; 58E30 -- 58A20 -- 37K05 -- 37K10; Nonlinear theories -- Periodicals; Mathematical physics -- Periodicals; 515.252
• DOI: 10.1080/14029251.2019.1640470
• Languages: English
• ISSNs: 1402-9251

Abstract

Abstract : For a scalar evolution equation ut = K ( t, x, u, ux, . . ., u 2 m +1 ) with m ≥ 1, the cohomology space H 1, 2 ( ) is shown to be isomorphic to the space of variational operators and an explicit isomorphism is given. The space of symplectic operators for ut = K for which the equation is Hamiltonian is also shown to be isomorphic to the space H 1, 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.