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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 $u_t-\varepsilon u_{xxt}+2\kappa u_x-u u_{xxx}-\alpha u u_x-\beta u_x u_{xx}=0$ with four arbitrary constants $$\varepsilon,\kappa,\alpha,\beta$$ are considered. This class has previously been studied by 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 $$\varepsilon,\kappa,\alpha,\beta$$. The computations are done with the help of the Maple package.

##### MSC:
 35A25 Other special methods applied to PDEs 58J70 Invariance and symmetry properties for PDEs on manifolds 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 35Q58 Other completely integrable PDE (MSC2000) 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 68W30 Symbolic computation and algebraic computation
##### Software:
diffgrob2; SYMMGRP.MAX; Maple; SYMMGRP
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