Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 486458, 18 pages doi:10.1155/2012/486458 Research Article New Traveling Wave Solutions by the Extended Generalized Riccati Equation Mapping Method of the 2 1-Dimensional Evolution Equation Hasibun Naher and Farah Aini Abdullah School of Mathematical Sciences, Universiti Sains Malaysia (USM), 11800 Penang, Malaysia Correspondence should be addressed to Hasibun Naher, hasibun06tasauf@gmail.com Received 19 September 2012; Accepted 14 October 2012 Academic Editor: Mohamed A. Abdou Copyright q 2012 H. Naher and F. A. Abdullah. 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 generalized Riccati equation mapping is extended with the basic G ′ /G-expansion method which is powerful and straightforward mathematical tool for solving nonlinear partial differential equations. In this paper, we construct twenty-seven traveling wave solutions for the 21-dimen- sional modified Zakharov-Kuznetsov equation by applying this method. Further, the auxiliary equation G ′ η w uGη vG 2 η is executed with arbitrary constant coefficients and called the generalized Riccati equation. The obtained solutions including solitons and periodic solutions are illustrated through the hyperbolic functions, the trigonometric functions, and the rational func- tions. In addition, it is worth declaring that one of our solutions is identical for special case with already established result which verifies our other solutions. Moreover, some of obtained solutions are depicted in the figures with the aid of Maple. 1. Introduction The study of analytical solutions for nonlinear partial differential equations PDEs has become more imperative and stimulating research fields in mathematical physics, engineer- ing sciences, and other technical arena 1–47. In the recent past, a wide range of methods have been developed to construct traveling wave solutions of nonlinear PDEs such as, the inverse scattering method 1, the Backlund transformation method 2, the Hirota bilinear transformation method 3, the bifurcation method 4, 5, the Jacobi elliptic function expan- sion method 6–8, the Weierstrass elliptic function method 9, the direct algebraic method 10, the homotopy perturbation method 11, 12, the Exp-function method 13–17, and others 18–28. Recently, Wang et al. 29 presented a widely used method, called the G ′ /G-expan- sion method to obtain traveling wave solutions for some nonlinear evolution equations NLEEs. Further, in this method, the second-order linear ordinary differential equation