### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 066, 11 pages      arXiv:1105.5303      https://doi.org/10.3842/SIGMA.2011.066
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”

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

Stephen C. Anco a, Sajid Ali b and Thomas Wolf a
a) Department of Mathematics, Brock University, St. Catharines, ON L2S 3A1 Canada
b) School of Electrical Engineering and Computer Sciences, National University of Sciences and Technology, H-12 Campus, Islamabad 44000, Pakistan

Received March 05, 2011, in final form July 03, 2011; Published online July 12, 2011; Typos in the solutions are corrected August 02, 2013

Abstract
A novel symmetry method for finding exact solutions to nonlinear PDEs is illustrated by applying it to a semilinear reaction-diffusion equation in multi-dimensions. The method uses a separation ansatz to solve an equivalent first-order group foliation system whose independent and dependent variables respectively consist of the invariants and differential invariants of a given one-dimensional group of point symmetries for the reaction-diffusion equation. With this group-foliation reduction method, solutions of the reaction-diffusion equation are obtained in an explicit form, including group-invariant similarity solutions and travelling-wave solutions, as well as dynamically interesting solutions that are not invariant under any of the point symmetries admitted by this equation.

Key words: semilinear heat equation; similarity reduction; exact solutions; group foliation; symmetry.

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