Research Article The Application of the exp(−Φ())-Expansion Method for Finding the Exact Solutions of Two Integrable Equations Naila Sajid and Ghazala Akram Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan Correspondence should be addressed to Ghazala Akram; toghazala2003@yahoo.com Received 3 August 2018; Revised 26 October 2018; Accepted 4 November 2018; Published 26 November 2018 Academic Editor: Maria L. Gandarias Copyright © 2018 Naila Sajid and Ghazala Akram. Tis 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. In this article, the exp(−Φ()) method is connected to search for new hyperbolic, periodic, and rational solutions of (1 + 1)- dimensional ffh-order nonlinear integrable equation and (2 + 1)-dimensional Date-Jimbo-Kashiwara-Miwa equation. Te obtained solutions consist of trigonometric, hyperbolic, rational functions and W-shaped soliton. Furthermore, 3D and 2D graphs are plotted by choosing the suitable values of the parameters involved. 1. Introduction Firstly, consider the (1+1)-dimensional ffh-order nonlinear integrable equation. In [1], Wazwaz proposed a new (1 + 1)- dimensional ffh-order nonlinear integrable equation of the form   −  − 4 (    )  − 4 (    )  = 0, (1) where  = (,) stands for wave propagation of physical quantity and subscripts represent partial diferentiation with respect to the given variable and obtained multiple soliton solutions using the simplifed Hirota’s method established by Hereman and Nuseir [2]. Furthermore, Yuan et al. [3] consider the (2 + 1)- dimensional Date-Jimbo-Kashiwara-Miwa equation. To study the (2 + 1)-dimensional Date-Jimbo-Kashiwara- Miwa equation, many researchers considered the following integrable equation:   + 4    + 2    + 6    +  − 2  = 0, (2) where  is the real function of the variables , , and . With the help of the Hirota method and auxiliary variables, the bilinear B¨ acklund transformation and N-soliton solutions are obtained. Nonlinear evolution equations (NLEEs) are one of the fastest developing zones of research in the feld of science and engineering, especially in mathematical biology, nonlinear optics, optical fber, fuid mechanics, solid state physics, biophysics, chemical physics, chemical kinetics, etc. Many efective methods have been proposed to solve the NLEEs, such as the Hirota method [1], Hereman-Nuseir method [2], inverse scattering transformation [4], Painlev´ e technique [5], B¨ acklund transformation [6], Darboux transformation [7, 8], Binary-Bell-polynomial scheme [9], frst integral method [10, 11], (  /)−expansion method [12], the exp(−Φ()) expansion method [13], Exp-function method [14], ansatz method [15], sine-Gordon expansion method [16, 17], the trial equation method [18, 19], homotopy asymptotic [20], and so on. Te present paper organized as follows: In Section 2, description of the exp(−Φ()) method for fnding the exact traveling wave solutions of NLEEs is presented. Section 3 illustrates the method to solve the (1 + 1)-dimensional ffh- order nonlinear integrable equation and (2 + 1)-dimensional Date-Jimbo-Kashiwara-Miwa equation. Results and discus- sion are presented in Section 4. Finally, in Section 5, some conclusions are given. Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 5191736, 10 pages https://doi.org/10.1155/2018/5191736