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Zbl 0879.35005
Clarkson, P.A.; Mansfield, E.L.; Priestley, T.J.
Symmetries of a class of nonlinear third-order partial differential equations.
(English)
[J] Math. Comput. Modelling 25, No.8-9, 195-212 (1997). ISSN 0895-7177

Symmetry reductions of the following class of nonlinear third-order partial differential equations $$u_t-\varepsilon u_{xxt}+2\kappa u_x-u u_{xxx}-\alpha u u_x-\beta u_x u_{xx}=0$$ with four arbitrary constants $\varepsilon,\kappa,\alpha,\beta$ are considered. This class has previously been studied by {\it C. Gilson} and {\it A. Pickering} [Phys. A, Math. Gen. 28, 2871-2888 (1995; Zbl 0830.35127)] using Painlevé theory. It contains as special cases the Fornberg-Whitham, the Rosenau-Hyman, and the Camassa-Holm equation. The authors apply besides the standard symmetry approach also the non-classical method of {\it G. W. Bluman} and {\it J. D. Cole} [J. Math. Mech. 18, 1025-1042, (1969; Zbl 0187.03502)]. Using the so-called differential Gröbner bases developed by one of the authors they obtain a symmetry classification of the parameters $\varepsilon,\kappa,\alpha,\beta$. The computations are done with the help of the Maple package.
[W.M.Seiler (Karlsruhe)]
MSC 2000:
*35A25 Other special methods (PDE)
58J70 Invariance and symmetry properties
13P10 Polynomial ideals, Groebner bases
35Q58 Other completely integrable PDE
37J35 Completely integrable systems, etc.
37K10 Completely integrable systems etc.
68W30 Symbolic computation and algebraic computation

Keywords: symmetry reduction; non-classical symmetry; Painlevé theory; computer algebra

Citations: Zbl 0830.35127; Zbl 0187.03502

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