Lie symmetries and invariant solutions of the shallow-water equation

In this paper we have investigated the Lie symmetries and similarity reductions of the new completely integrable dispersive shallow water equation, discussed recently by Camassa and Holm, through classical and non-classical methods. We compare the similarity reduction thus obtained by the direct method of Clarkson and Kruskal. The resultant ordinary differential equation satisfies the Painleve property and the ARS conjecture. We also point out how the similarity reduction of the present integrable system differs from that of the non-integrable, but closely related, BBM equation. (C) 1996 Elsevier Science Ltd.