Mixed superposition rules and the Riccati hierarchy
Abstract
Mixed superposition rules, i.e., functions describing the general solution of a system of first-order differential equations in terms of a generic family of particular solutions of first-order systems and some constants, are studied. The main achievement is a generalization of the celebrated Lie-Scheffers Theorem, characterizing systems admitting a mixed superposition rule. This somehow unexpected result says that such systems are exactly Lie systems, i.e., they admit a standard superposition rule. This provides a new and powerful tool for finding Lie systems, which is applied here to studying the Riccati hierarchy and to retrieving some known results in a more efficient and simpler way.
- Publication:
-
Journal of Differential Equations
- Pub Date:
- 2013
- DOI:
- 10.1016/j.jde.2012.08.020
- arXiv:
- arXiv:1203.0123
- Bibcode:
- 2013JDE...254..179G
- Keywords:
-
- Mathematics - Classical Analysis and ODEs;
- Mathematical Physics;
- Mathematics - Differential Geometry;
- 34A26 (Primary);
- 53B50;
- 17C90;
- (Secondary)
- E-Print:
- 20 pages