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International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 2, Pages 263-277

On the Korteweg-de Vries equation: an associated equation

1Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga 37402, Tennessee, USA
2Department of Systems Science and Mathematics, Washington University, St. Louis 63130, Missouri, USA

Received 12 December 1983

Copyright © 1984 Hindawi Publishing Corporation. 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 purpose of this paper is to describe a relationship between the Korteweg-de Vries (KdV) equation ut6uux+uxxx=0and another nonlinear partial differential equation of the form zt+zxxx3zxzxxz=H(t)z.The second equation will be called the Associated Equation (AE) and the connection between the two will be explained. By considering AE, explicit solutions to KdV will be obtained. These solutions include the solitary wave and the cnoidal wave solutions. In addition, similarity solutions in terms of Airy functions and Painlevé transcendents are found. The approach here is different from the Inverse Scattering Transform and the results are not in the form of solutions to specific initial value problems, but rather in terms of solutions containing arbitrary constants.