The Riccati and Ermakov-Pinney hierarchies

DOI

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.

Copyright

© 2007, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

Download article (PDF)

TY  - JOUR
AU  - Marianna Euler
AU  - Norbert Euler
AU  - Peter Leach
PY  - 2007
DA  - 2007/04/01
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  - 10.2991/jnmp.2007.14.2.10
ID  - Euler2007
ER  -