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Mathematical Problems in Engineering
Volume 2012, Article ID 725061, 14 pages
http://dx.doi.org/10.1155/2012/725061
Research Article

The Two-Variable ( ๐บ โ€ฒ / ๐บ , 1 / ๐บ ) -Expansion Method for Solving the Nonlinear KdV-mKdV Equation

Mathematics Department, Faculty of Science, Zagazig University, Zagazig 44519, Egypt

Received 6 January 2012; Accepted 17 March 2012

Academic Editor: Gradimir V. Milovanoviฤ‡

Copyright © 2012 E. M. E. Zayed and M. A. M. Abdelaziz. 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.

Abstract

We apply the two-variable ( ๐บ ๎…ž / ๐บ , 1 / ๐บ )-expansion method to construct new exact traveling wave solutions with parameters of the nonlinear ( 1 + 1 )-dimensional KdV-mKdV equation. This method can be thought of as the generalization of the well-known ( ๐บ ๎…ž / ๐บ )-expansion method given recently by M. Wang et al. When the parameters are replaced by special values, the well-known solitary wave solutions of this equation are rediscovered from the traveling waves. It is shown that the proposed method provides a more general powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.