Journal of Nonlinear Mathematical Physics

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
https://doi.org/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() 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() 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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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
26 - 4
Pages
604 - 649
Publication Date
2019/07
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/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

TY  - JOUR
AU  - M.E. Fels
AU  - E. Yaşar
PY  - 2019
DA  - 2019/07
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  - https://doi.org/10.1080/14029251.2019.1640470
ID  - Fels2019
ER  -