Lie systems, Lie symmetries and reciprocal transformations

TL;DR

This thesis explores Lie systems with geometric structures, Lie symmetries of differential equations, and reciprocal transformations, highlighting their applications in mathematical physics, biology, and mathematics.

Contribution

It introduces new Lie systems with Hamiltonian structures, analyzes Lie symmetries of hierarchies like Camassa-Holm, and studies reciprocal transformations in physical differential equations.

Findings

New Lie systems with Hamiltonian vector fields identified

Lie symmetries of integrable hierarchies analyzed

Reciprocal transformations applied to key physical equations

Abstract

This work represents a PhD thesis concerning three main topics. The first one deals with the study and applications of Lie systems with compatible geometric structures, e.g. symplectic, Poisson, Dirac, Jacobi, among others. Many new Lie systems admitting Vessiot--Guldberg Lie algebras of Hamiltonian vector fields relative to the above mentioned geometric structures are analyzed and their importance is illustrated by their appearances in physical, biological and mathematical models. The second part details the study of Lie symmetries and reductions of relevant hierarchies of differential equations and their corresponding Lax pairs. For example, the Cammasa-Holm and Qiao hierarchies in 2+1 dimensions. The third and last part is dedicated to the study of reciprocal transformations and their application in differential equations appearing in mathematical physics, e.g. Qiao and Camassa-Holm equations equations appearing in the second part.