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Abstract and Applied Analysis
Volume 2013, Article ID 109690, 7 pages
Research Article

Abundant Explicit and Exact Solutions for the Variable Coefficient mKdV Equations

1School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong 510006, China
2Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong, Higher Education Institutes, Guangzhou University, Guangzhou, Guangdong 510006, China
3School of Computer Science and Educational Software, Guangzhou University, Guangzhou, Guangdong 510006, China

Received 30 October 2013; Accepted 7 December 2013

Academic Editor: Dianchen Lu

Copyright © 2013 Xiaoxiao Zheng et al. 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.


This paper is concerned with the variable coefficients mKdV (VC-mKdV) equation. First, through some transformation we convert VC-mKdV equation into the constant coefficient mKdV equation. Then, using the first integral method we obtain the exact solutions of VC-mKdV equation, such as rational function solutions, periodic wave solutions of triangle function, bell-shape solitary wave solution, kink-shape solitary wave solution, Jacobi elliptic function solutions, and Weierstrass elliptic function solution. Furthermore, with the aid of Mathematica, the extended hyperbolic functions method is used to establish abundant exact explicit solution of VC-mKdV equation. By the results of the equation, the first integral method and the extended hyperbolic function method are extended from the constant coefficient nonlinear evolution equations to the variable coefficients nonlinear partial differential equation.