Geometric Models for Lie–Hamilton Systems on ℝ2

This paper provides a geometric description for Lie–Hamilton systems on R 2 with locally transitive Vessiot–Guldberg Lie algebras through two types of geometric models. The first one is the restriction of a class of Lie–Hamilton systems on the dual of a Lie algebra to even-dimensional symplectic leaves relative to the Kirillov-Kostant-Souriau bracket. The second is a projection onto a quotient space of an automorphic Lie–Hamilton system relative to a naturally defined Poisson structure or, more generally, an automorphic Lie system with a compatible bivector field. These models give a natural framework for the analysis of Lie–Hamilton systems on R 2 while retrieving known results in a natural manner. Our methods may be extended to study Lie–Hamilton systems on higher-dimensional manifolds and provide new approaches to Lie systems admitting compatible geometric structures.

Quasi-Lie schemes for PDEs

The theory of quasi-Lie systems, i.e. systems of first-order ordinary differential equations that can be related via a generalized flow to Lie systems, is extended to systems of partial differential equations (PDEs) and its applications to obtain [Formula: see text]-dependent superposition rules, and integrability conditions are analyzed. We develop a procedure of constructing quasi-Lie systems through a generalization to PDEs of the so-called theory of quasi-Lie schemes. Our techniques are illustrated with the analysis of Wess–Zumino–Novikov–Witten models, generalized Abel differential equations, Bäcklund transformations, as well as other differential equations of physical and mathematical relevance.