Volume 14, Issue 2, April 2007, Pages 290 - 310
The Riccati and Ermakov-Pinney hierarchies
- Marianna EULER, Norbert EULER, Peter LEACH
- Corresponding Author
- Marianna EULER
Available Online 1 April 2007.
- https://doi.org/10.2991/jnmp.2007.14.2.10How to use a DOI?
- 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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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 -