Journal of Nonlinear Mathematical Physics

Volume 16, Issue Supplement 1, March 2013, Pages 137 - 147

Integration of Systems of First-Order Equations Admitting Nonlinear Superposition

Authors
N. H. Ibragimov
Department of Mathematics and Science, Research Centre ALGA: Advances in Lie Group Analysis, Blekinge Institute of Technology, SE-371 79 Karlskrona, Sweden
Received 13 June 2009, Accepted 7 September 2009, Available Online 7 January 2021.
DOI
https://doi.org/10.1142/S1402925109000364How to use a DOI?
Keywords
Nonlinear superposition, Vessiot–Guldberg–Lie algebras, standard forms of Lie algebras, canonical variables
Abstract

Systems of two nonlinear ordinary differential equations of the first order admitting nonlinear superpositions are investigated using Lie’s enumeration of groups on the plane. It is shown that the systems associated with two-dimensional Vessiot–Guldberg–Lie algebras can be integrated by quadrature upon introducing Lie’s canonical variables. The knowledge of a symmetry group of a system in question is not needed in this approach. The systems associated with three-dimensional Vessiot–Guldberg–Lie algebras are classified into 13 standard forms 10 of which are integrable by quadratures and three are reduced to Riccati equations.

Copyright
© 2009 The Authors. Published by Atlantis Press 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
16 - Supplement 1
Pages
137 - 147
Publication Date
2021/01
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.1142/S1402925109000364How to use a DOI?
Copyright
© 2009 The Authors. Published by Atlantis Press 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  - N. H. Ibragimov
PY  - 2021
DA  - 2021/01
TI  - Integration of Systems of First-Order Equations Admitting Nonlinear Superposition
JO  - Journal of Nonlinear Mathematical Physics
SP  - 137
EP  - 147
VL  - 16
IS  - Supplement 1
SN  - 1776-0852
UR  - https://doi.org/10.1142/S1402925109000364
DO  - https://doi.org/10.1142/S1402925109000364
ID  - Ibragimov2021
ER  -