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

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

2011-05-26

arxiv.org/abs/1105.5303v3

SIGMA 7 (2011), 066, 10 pages

Physics

Mathematical Physics

Scientific paper

10.3842/SIGMA.2011.066

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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