Geometric Approach to Invariance Groups and Solution of Partial Differential Systems

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Abstract
Methods are discussed for discovery of physically or mathematically special families of exact solutions of systems of partial differential equations. Such systems are described geometrically using equivalent sets of differential forms, and the theory derived for obtaining the generators of their invariance groups‐vector fields in the space of forms. These isovectors then lead naturally to all the special solutions discussed, and it appears that other special ansätze must similarly be capable of geometric description. Application is made to the one‐dimensional heat equation, the vacuum Maxwell equations, the Korteweg‐de Vries equation, one‐dimensional compressible fluid dynamics, the Lambropoulos equation, and the cylindrically symmetric Einstein‐Maxwell equations.