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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

${u}_{t}-\epsilon {u}_{xxt}+2\kappa {u}_{x}-u{u}_{xxx}-\alpha u{u}_{x}-\beta {u}_{x}{u}_{xx}=0$

with four arbitrary constants $\epsilon ,\kappa ,\alpha ,\beta$ 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 $\epsilon ,\kappa ,\alpha ,\beta$. The computations are done with the help of the Maple package.

##### MSC:
 35A25 Other special methods (PDE) 58J70 Invariance and symmetry properties 13P10 Gröbner bases; other bases for ideals and modules 35Q58 Other completely integrable PDE (MSC2000) 37J35 Completely integrable systems, topological structure of phase space, integration methods 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies 68W30 Symbolic computation and algebraic computation
SYMMGRP; Maple