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A novel symmetry method for finding exact solutions to nonlinear PDEs isillustrated by applying it to a semilinear reaction-diffusion equation inmulti-dimensions. The method uses a separation ansatz to solve an equivalentfirst-order group foliation system whose independent and dependent variablesrespectively consist of the invariants and differential invariants of a givenone-dimensional group of point symmetries for the reaction-diffusion equation.With this group-foliation reduction method, solutions of the reaction-diffusionequation are obtained in an explicit form, including group-invariant similaritysolutions and travelling-wave solutions, as well as dynamically interestingsolutions that are not invariant under any of the point symmetries admitted bythis equation.

Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction