A Lie systems approach to the Riccati hierarchy and partial differential equations

TL;DR

This paper uses Lie systems to unify and analyze the Riccati hierarchy, revealing geometric structures and deriving superposition rules, with applications to integrable PDEs like Kaup-Kupershmidt and Sawada-Kotera.

Contribution

It introduces a Lie systems framework to study Riccati chain equations as projective Riccati equations, simplifying their analysis and linking them to integrable PDEs.

Findings

Riccati hierarchy can be viewed as projective Riccati equations.

Superposition rules depend on fewer solutions for certain cases.

Applications to integrable PDEs like Kaup-Kupershmidt and Sawada-Kotera.

Abstract

It is proved that the members of the Riccati hierarchy, the so-called Riccati chain equations, can be considered as particular cases of projective Riccati equations, which greatly simplifies the study of the Riccati hierarchy. This also allows us to characterize Riccati chain equations geometrically in terms of the projective vector fields of a flat Riemannian metric and to easily derive their associated superposition rules. Next, we establish necessary and sufficient conditions under which it is possible to map second-order Riccati chain equations into conformal Riccati equations through a local diffeomorphism. This fact can be used to determine superposition rules for particular higher-order Riccati chain equations which depend on fewer particular solutions than in the general case. Therefore, we analyze the properties of Euclidean, hyperbolic and projective vector fields on the plane in detail. Finally, the use of contact transformations enables us to apply the derived results to the study of certain integrable partial differential equations, such as the Kaup-Kupershmidt and Sawada-Kotera equations.