Journal of Nonlinear Mathematical Physics

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Volume 26, Issue 4, July 2019, Pages 604 - 649

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

Authors
M.E. Fels
Department of Mathematics and Statistics, Utah State University, Logan Utah, 84322, USA,mark.fels@usu.edu
E. Yaşar
Department of Mathematics, Uludag University, Bursa, Turkey,eyasar@uludag.edu.tr
Received 13 March 2019, Accepted 4 June 2019, Available Online 9 July 2019.
DOI
10.1080/14029251.2019.1640470How to use a DOI?
Keywords
Variational Bicomplex; Cohomology; Scalar Evolution Equation; Symplectic Operator; Hamiltonian Evolution Equation
Abstract

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.

Copyright
© 2019 The Authors. Published by Atlantis and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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<Previous Article In Issue
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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
26 - 4
Pages
604 - 649
Publication Date
2019/07/09
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1080/14029251.2019.1640470How to use a DOI?
Copyright
© 2019 The Authors. Published by Atlantis and Taylor & Francis
Open Access
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

risenwbib
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  -