A contribution to the symmetry classification problem for second-order PDEs in z(x, y)

Abstract

I report on a contribution to the point symmetry classification problem for second-order partial differential equations (PDEs) in z(x, y), i.e. to an overview over all possible symmetry groups admitted by this class of equations. The article also contains a concise introduction into classical symmetry analysis. Sophus Lie (1842-1899) determined all continuous transformation groups of the 2D plane and gave normal forms for any ordinary differential equation that is invariant under one of those groups. I deal with the extension of Lie's program to second-order PDEs in z(x, y). The starting point to this endeavour is a previously unknown paper by Amaldi from 1901, which claims to have completed Lie's classification of groups acting in (x, y, z)-space. T also present a Maple procedure ('LHSO1_PDE_Solver') for solving systems of linear, homogeneous first-order PDEs that performs better on this class than Maple's built-in PDE system solver.