A Point Symmetry Group of a Differential Equation which cannot be Found Using Infinitesimal Methods

Abstract

A point symmetry group of a differential equation is a group of invertible transformations of its dependent and independent variables which leaves the differential equation invariant. Lie’s infinitesimal method recovers the connected components of Lie symmetry groups of differential equations, and, by analytic continuation, sometimes yields disconnected components.

We show that Lie’s infinitesimal method and its analytic continuation fails to yield the full group of point symmetries of the ODE ${u_{{xx}}} = \frac{1}{x}{u_x} + \frac{4}{{{x^3}}}{u^2}$. To find the full point symmetry group of this ODE we apply a differential analogue of Buchberger’s algorithm to simplify its nonlinear determining equations. The reduced form of these equations is easily solved to yield the explicit form of the group transformations.