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Symmetries of a class of nonlinear third-order partial differential equations. (English) Zbl 0879.35005

Symmetry reductions of the following class of nonlinear third-order partial differential equations

u t -εu xxt +2κu x -uu xxx -αuu x -βu x u xx =0

with four arbitrary constants ε,κ,α,β are considered. This class has previously been studied by C. Gilson and 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 G. W. Bluman and 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 ε,κ,α,β. The computations are done with the help of the Maple package.

35A25Other special methods (PDE)
58J70Invariance and symmetry properties
13P10Gröbner bases; other bases for ideals and modules
35Q58Other completely integrable PDE (MSC2000)
37J35Completely integrable systems, topological structure of phase space, integration methods
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
68W30Symbolic computation and algebraic computation