International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 13, Pages 2159-2173

Symmetry properties of a generalized Korteweg-de Vries equation and some explicit solutions

Paul Bracken

Department of Mathematics, University of Texas-Pan American, Edinburg 78539-2999, TX, USA

Received 18 October 2004

Copyright © 2005 Paul Bracken. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The symmetry group method is applied to a generalized Korteweg-de Vries equation and several classes of group invariant solutions for it are obtained by means of this technique. Polynomial, trigonometric, and elliptic function solutions can be calculated. It is shown that this generalized equation can be reduced to a first-order equation under a particular second-order differential constraint which resembles a Schrödinger equation. For a particular instance in which the constraint is satisfied, the generalized equation is reduced to a quadrature. A condition which ensures that the reciprocal of a solution is also a solution is given, and a first integral to this constraint is found.