UBC Theses and Dissertations
New symmetries from old: exploiting lie algebra structure to determine infinitesimal symmetries of differential equations Boulton, Alan
We give a method for using explicitly known Lie symmetries of a system of differential equations to help find more symmetries of the system. A Lie (or infinitesimal) symmetry of a system of differential equations is a transformation of its dependent and independent variables, depending on continuous parameters, which maps any solution of the system to another solution of the same system. Infinitesimal Lie symmetries of a system of differential equations arise as solutions of a related system of linear homogeneous partial differential equations called infinitesimal determining equations. The importance of symmetries in applications has prompted the development of many software packages to derive and attempt to integrate infinitesimal determining equations. For a given system of differential equations we usually have a priori explicit knowledge of many symmetries of the system because of their simple form or the physical origin of the system. Current methods for finding symmetries do not incorporate this a priori information. Our method uses such information to simplify the problem of finding the remaining unknown symmetries by exploiting the Lie algebra structure of the solution space of the infinitesimal determining equations. We illustrate our method for simplifying infinitesimal determining systems by applying it to three well known test problems: the linear heat equation; Laplace's equation; and a class of nonlinear diffusion equations. Our method uses the inspectional symmetries of each of these differential equations to determine the equation's remaining symmetries. In these cases the method is so effective that the simplified determining systems for the unknown symmetries are just linear (algebraic) equations. Finally we indicate that it is possible to obtain determining equations for the infinitesimal generators of various subalgebras of a Lie symmetry algebra specified by infinitesimal determining equations. In particular we prove that the determining equations for the centre of the algebra can always be obtained.
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