Variational Operators, Symplectic Operators, and the Cohomology of Scalar Evolution Equations
- 10.1080/14029251.2019.1640470How to use a DOI?
- Variational Bicomplex; Cohomology; Scalar Evolution Equation; Symplectic Operator; Hamiltonian Evolution Equation
For a scalar evolution equation ut = K(t, x, u, ux, ..., u2m+1) with m ≥ 1, the cohomology space H1,2(ℛ∞) 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 H1,2(ℛ∞) 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.
- © 2019 The Authors. Published by Atlantis and Taylor & Francis
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- This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).
Cite this article
TY - JOUR AU - M.E. Fels AU - E. Yaşar PY - 2019 DA - 2019/07/09 TI - Variational Operators, Symplectic Operators, and the Cohomology of Scalar Evolution Equations JO - Journal of Nonlinear Mathematical Physics SP - 604 EP - 649 VL - 26 IS - 4 SN - 1776-0852 UR - https://doi.org/10.1080/14029251.2019.1640470 DO - 10.1080/14029251.2019.1640470 ID - Fels2019 ER -