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

Please always quote using this URN: urn:nbn:de:0297-zib-13171
  • 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 is based on group foliation reduction and employs 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 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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Author:Stephen S. Anco, S. Ali, Thomas Wolf
Document Type:ZIB-Report
Tag:semilinear heat equation, similarity reduction, exact solutions, group foliation, symmetry
MSC-Classification:34-XX ORDINARY DIFFERENTIAL EQUATIONS
35-XX PARTIAL DIFFERENTIAL EQUATIONS
58-XX GLOBAL ANALYSIS, ANALYSIS ON MANIFOLDS [See also 32Cxx, 32Fxx, 32Wxx, 46-XX, 47Hxx, 53Cxx](For geometric integration theory, see 49Q15)
Date of first Publication:2011/06/21
Series (Serial Number):ZIB-Report (11-25)
Published in:Appeared in: SIGMA 7 (2011), 066, 10 pages.