Journal of Nonlinear Mathematical Physics

Volume 14, Issue 2, April 2007, Pages 290 - 310

The Riccati and Ermakov-Pinney hierarchies

Authors
Marianna EULER, Norbert EULER, Peter LEACH
Corresponding Author
Marianna EULER
Available Online 1 April 2007.
DOI
https://doi.org/10.2991/jnmp.2007.14.2.10How to use a DOI?
Abstract
The concept and use of recursion operators is well-established in the study of evolution, in particular nonlinear, equations. We demonstrate the application of the idea of recursion operators to ordinary differential equations. For the purposes of our demonstration we use two equations, one chosen from the class of linearisable hierarchies of evolution equations studied by Euler et al (Stud Appl Math 111 (2003) 315-337) and the other from the class of integrable but nonlinearisible equations studied by Petersson et al (Stud Appl Math 112 (2004) 201-225). We construct the hierarchies for each equation. The symmetry properties of the first hierarchy are considered in some detail. For both hierarchies we apply the singularity analysis. For both we observe intersting behaviour of the resonances for the different possible leading order behaviours. In particular we note the proliferation of subsidiary solutions as one ascends the hierarchy.
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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
14 - 2
Pages
290 - 310
Publication Date
2007/04
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.2991/jnmp.2007.14.2.10How to use a DOI?
Open Access
This is an open access article distributed under the CC BY-NC license.

Cite this article

TY  - JOUR
AU  - Marianna EULER
AU  - Norbert EULER
AU  - Peter LEACH
PY  - 2007
DA  - 2007/04
TI  - The Riccati and Ermakov-Pinney hierarchies
JO  - Journal of Nonlinear Mathematical Physics
SP  - 290
EP  - 310
VL  - 14
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2007.14.2.10
DO  - https://doi.org/10.2991/jnmp.2007.14.2.10
ID  - EULER2007
ER  -