On Soliton Solutions of the Drinfeld-Sokolov- Wilson System by He’s Variational Principle Mohammed K Elboree* Department of Mathematics, Egypt Abstract The aim of this paper is to obtain the traveling wave solutions for the Drinfeld-Sokolov-Wilson system by He’s semi-inverse variational principle which includes the solitary and periodic wave solutions by a suitable choice for the parameters. We analysis these solutions physically by some gores to complement this study. Finally, this method can be used successfully for solving integrable and nonintegrable equations. Keywords: He’s semi-inverse variational principle; Traveling wave solutions; Dronfield-Sokolov-Wilson system; Soliton solutions; Periodic wave solutions Introduction Nonlinear partial differential equations (NLPDEs) is used to describe many important phenomena in mathematical physics, mechanics, chemistry, biology, etc. Such as Cortège de Vries equation, Burgers equation, Schrodinger equation, Bossiness equation and so on. So, the discovery of the exact solutions of nonlinear partial differential equations is of the most important priorities. Many effective methods are used to construct traveling wave solutions of NLPDEs, among of these methods, Adomian decomposition method [1], the homotropy perturbation method [2], the variational iteration method [3, 4], the He’s variational approach [5], the extended homoclinic test approach [6, 7], homogeneous balance method [8-11], Jacobi elliptic function method [12-15], Baclund transformation [16, 17], G 0 =G expansion method [18] . He’s semi-inverse variational principle is used to obtain the traveling wave solutions for Dronfield-Sokolov-Wilson system which include the solitary and periodic wave solutions by a suitable choice for the parameters. It is important to point out that a new constrained variational principle for heat conduction is obtained recently via the semi-inverse method combined with separation of variables[19], which is an exactly the same with He-Lee’s variational principle[20] a short remark on the history of the semi-inverse method for establishment of a generalized variational principle is given in [21]. In soliton theory, we aim to search for the solitary wave solutions for NLPDEs using various methods [22]. In particular, we used in this paper the variational principle, which enables us to and the Lagrangian for the Drinfeld-Sokolov-Wilson system, which related to the conservation laws which plays an important role in solution process [23,24] and provides physical insight into the nature of the solution of this problem as shown by gures. Also, this method helps in establishing connections between the physics and mathematics and much more active than the Noether’s theorem [25, 26]. The key idea in this paper is to use the He’s semi-inverse variational principle to and out several exact solutions for Drinfeld-Sokolov-Wilson system. Conclusions is given in the last of this paper. Methodology Suppose we are given a nonlinear partial differential equations (NLPDEs) in the following form N (u; u ; u ; u ; ::::) =0; t x xt (1) where x and t are the independent variables. This method can be summarizing as follows Crimson Publishers Wings to the Research Research Article *Corresponding author: Mohammed K Elboree, Department of Mathematics, Egypt Submission: November 29, 2018 Published: April 4, 2019 Volume 2 - Issue 4 How to cite this article: Mohammed K E. On Soliton Solutions of the Drinfeld- Sokolov-Wilson System by He’s Variational Principle. Evolutions Mech Eng.2(4). EME.000543.2019. DOI: 10.31031/EME.2019.02.000543 Copyright@ Mohammed K Elboree, This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use and redistribution provided that the original author and source are credited. ISSN: 2640-9690 1 Evolutions in Mechanical Engineering