Research Article Classification of Multiply Travelling Wave Solutions for Coupled Burgers, Combined KdV-Modified KdV, and Schrödinger-KdV Equations A. R. Seadawy 1,2 and K. El-Rashidy 2,3 1 Mathematics Department, Faculty of Science, Taibah University, Al-Ula, Saudi Arabia 2 Mathematics Department, Faculty of Science, Beni-Suef University, Egypt 3 Mathematics Department, College of Arts and Science, Taif University, Ranyah, Saudi Arabia Correspondence should be addressed to A. R. Seadawy; aly742001@yahoo.com Received 24 August 2014; Accepted 1 November 2014 Academic Editor: Yasir Khan Copyright © 2015 A. R. Seadawy and K. El-Rashidy. his 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. Some explicit travelling wave solutions to constructing exact solutions of nonlinear partial diferential equations of mathematical physics are presented. By applying a theory of Frobenius decompositions and, more precisely, by using a transformation method to the coupled Burgers, combined Korteweg-de Vries- (KdV-) modiied KdV and Schr¨ odinger-KdV equation is written as bilinear ordinary diferential equations and two solutions to describing nonlinear interaction of travelling waves are generated. he properties of the multiple travelling wave solutions are shown by some igures. All solutions are stable and have applications in physics. 1. Introduction he investigation of traveling wave solutions of nonlinear evolution equations (NLEEs) plays a vital role in diferent branches of mathematical physics, engineering sciences, and other technical arenas, such as plasma physics, nonlinear optics, solid state physics, luid mechanics, chemical physics, and chemistry. he Burgers’ equation has been found to describe various kinds of phenomena such as a mathematical model of turbulence [1] and the approximate theory of low through a shock wave traveling in viscous luid [2]. Fletcher using the Hopf-Cole transformation [3] gave an analytic solution for the system of two-dimensional Burgers’ equations. he Korteweg-de Vries (KdV) equation which models shallow-water phenomena has been analyzed extensively using the invariance properties that occur from the Lie point symmetry generator that admits it. In particular, travelling wave solutions arise from the combination of translations in space and time. Also, Galilean invariants and scale-invariant solutions are dependent on irst and second Painleve transcendent [4]. Further, the modiied KdV (mKdV) has attracted interest in a similar way and its Lie point symmetry generators are known [5]. Recently, the combined KdV (cKdV) and mKdV equation has been studied using various methods with a special reference to soliton- type solutions. For example, simple soliton solutions to cKdV- mKdV used in plasma and luid physics are obtained in [6]. Here, the particular form uses the fact that the equation admits a scaling symmetry which is nonexistent for the general cKdV [7, 8]. he topic of solitons produced by nonlinear interactions is a very fundamental topic in various ields, including optical solitons in ibers [9]. he one-dimensional soliton can be considered as a localized wave pulse that propagates along one space direction undeformed; that is, dispersion is completely compensated by the nonlinear efects. here is an enormous amount of literature about the integrability of nonlinear equations related to scattering equations, including especially inverse scattering theories, in relation to solitons [10]. In particular, analysis related to NLS and KdV equations has been studied [10–12]. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2015, Article ID 369294, 7 pages http://dx.doi.org/10.1155/2015/369294