Quasi-rectifiable Lie algebras for partial differential equations

Abstract

We introduce families of quasi-rectifiable vector fields and study their geometric and algebraic aspects. Then, we analyse their applications to systems of partial differential equations. Our results explain, in a simpler manner, previous findings about hydrodynamic-type equations. Facts concerning families of quasi-rectifiable vector fields, their relation to Hamiltonian systems, and practical procedures for studying such families are developed. We introduce and analyse quasi-rectifiable Lie algebras, which are motivated by geometric and practical reasons. We classify different types of quasi-rectifiable Lie algebras, e.g. indecomposable ones up to dimension five. New methods for solving systems of hydrodynamic-type equations are established to illustrate our results. In particular, we study hydrodynamic-type systems admitting $k$-wave solutions through quasi-rectifiable Lie algebras of vector fields. We develop techniques for obtaining the submanifolds related to quasi-rectifiable Lie algebras of vector fields and systems of partial differential equations admitting a nonlinear superposition rule: the PDE Lie systems.