Mixed superposition rules and the Riccati hierarchy

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:254

Article

Grabowski, Janusz¹  de Lucas, Javier¹ 

[1] Polish Acad Sci, Inst Math, PL-00956 Warsaw, Poland

关键词: Lie system;    Superposition rule;    Mixed superposition rule;    Vessiot-Guldberg Lie algebra;    Riccati hierarchy;    Milne-Pinney equation;    Kummer-Schwarz equation;   

DOI  :  10.1016/j.jde.2012.08.020

来源: Elsevier

【 摘 要 】

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. (c) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

Free