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Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2011, том 7
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Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction
Інший ID:
2010 Mathematics Subject Classification: 35K58; 35C06; 35A25; 58J70; 34C14
DOI:10.3842/SIGMA.2011.066
DOI:10.3842/SIGMA.2011.066
Посилання:
Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction / S.C. Anco, S. Ali, T. Wolf // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 16 назв. — англ.
Підтримка:
This paper is a contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S⁴)”. The full collection is available at http://www.emis.de/journals/SIGMA/S4.html.
S. Anco and T. Wolf are each supported by an NSERC research grant. S. Ali thanks the
Mathematics Department of Brock University for support during the period of a research visit when this paper was written. Computations were partly performed on computers of the Sharcnet consortium (www.sharcnet.ca). The referees and the editor are thanked for valuable comments which have improved this paper.
Дата:
2011
Переглядів:
389
Завантажень:
235
Анотація:
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.
