Lie and Noether point symmetries of a class of quasilinear systems of second-order differential equations

Abstract

We study the Lie and Noether point symmetries of a class of systems of second-order differential equations with $n$ independent and $m$ dependent variables ($n\times m$ systems). We solve the symmetry conditions in a geometric way and determine the general form of the symmetry vector and of the Noetherian conservation laws. We prove that the point symmetries are generated by the collineations of two (pseudo)metrics, which are defined in the spaces of independent and dependent variables. We demonstrate the general results in two special cases (a) a system of $m$ coupled Laplace equations and (b) the Klein-Gordon equation of a particle in the context of Generalized Uncertainty Principle. In the second case we determine the complete invariant group of point transformations, and we apply the Lie invariants in order to find invariant solutions of the wave function for a spin-$0$ particle in the two dimensional hyperbolic space.

20 pages, to be published in Journal of Geometry and Physics

References in corpus (4)

• Universality of Quantum Gravity Corrections
• Lie and Noether symmetries of geodesic equations and collineations
• Two dimensional dynamical systems which admit Lie and Noether symmetries
• Lie point symmetries of a general class of PDEs: The heat equation

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• Lie symmetries for systems of evolution equations
• Constructing the CKVs of Bianchi III and V spacetimes
• Modified Brans-Dicke cosmology with Minimum Length Uncertainty
• The Unified Soliton System as the $\mathrm{Ad}{\mathrm{S}}_2$ System