Quasi-rectifiable Lie algebras for partial differential equations
arXiv:2312.05238 · doi:10.1088/1361-6544/ada50e
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 -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.
45 pages. Improved terminology. Typos and minor issues corrected
References in corpus (8)
- Superposition rules, Lie theorem and partial differential equations
- Poisson-Hopf algebra deformations of Lie-Hamilton systems
- Contact Lie systems
- Sundman transformation and alternative tangent structuresSundman transformation and alternative tangent structures
- Infinitesimal time reparametrisation and its applications
- Multiple Riemann wave solutions of the general form of quasilinear hyperbolic systems
- Generalized time-dependent SIS Hamiltonian models: Exact solutions and quantum deformations
- On k-wave solutions of quasilinear systems of partial differential equations