Exact solutions of nonlinear partial differential equations by the method of group foliation reduction

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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.

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Anco, S. C., Ali, S., & Wolf, T. (2011). Exact solutions of nonlinear partial differential equations by the method of group foliation reduction. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 7. https://doi.org/10.3842/SIGMA.2011.066

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