Electron. J. Diff. Equ.,
Vol. 2015 (2015), No. 310, pp. 17.
Fields of rational constants of cyclic factorizable derivations
Janusz Zielinski
Abstract:
We describe all rational constants of a large family of fourvariable
cyclic factorizable derivations. Thus, we determine all rational
first integrals of their corresponding systems of differential equations.
Moreover, we give a characteristic of all fourvariable LotkaVolterra
derivations with a nontrivial rational constant.
All considerations are over an arbitrary field of characteristic zero.
Our main tool is the investigation of the cofactors of strict Darboux
polynomials. Factorizable derivations are important in derivation theory.
Namely, we may associate the factorizable derivation with any given
derivation of a polynomial ring and that construction
helps to determine rational constants of arbitrary derivations.
Besides, LotkaVolterra systems play a significant role in population
biology, laser physics and plasma physics.
Submitted November 12, 2014. Published December 21, 2015.
Math Subject Classifications: 34A34, 13N15, 12H05, 92D25.
Key Words: LotkaVolterra derivation; factorizable derivation;
rational constant; rational first integral.
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Janusz Zielinski
Faculty of Mathematics and Computer Science
N. Copernicus University, ul. Chopina 12/18
87100 Torun, Poland
email: ubukrool@mat.uni.torun.pl

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