Eur. Phys. J. Plus (2017) 132: 155
https://doi.org/10.1140/epjp/i2017-11463-3

Regular Article

Applications of He's semi-inverse method, ITEM and GGM to the Davey-Stewartson equation

¹ Department of Mechanical Engineering, Tabriz Branch, Islamic Azad University, Tabriz, Iran
² Young Researchers and Elite Club, Ilkhchi Branch, Islamic Azad University, Ilkhchi, Iran

^* e-mail: j_manafianheris@tabrizu.ac.ir

Received: 26 January 2017
Accepted: 5 March 2017
Published online: 10 April 2017

Abstract

We investigate the Davey-Stewartson (DS) equation. Travelling wave solutions were found. In this paper, we demonstrate the effectiveness of the analytical methods, namely, He's semi-inverse variational principle method (SIVPM), the improved -expansion method (ITEM) and generalized -expansion method (GGM) for seeking more exact solutions via the DS equation. These methods are direct, concise and simple to implement compared to other existing methods. The exact solutions containing four types solutions have been achieved. The results demonstrate that the aforementioned methods are more efficient than the Ansatz method applied by Mirzazadeh (2015). Abundant exact travelling wave solutions including solitons, kink, periodic and rational solutions have been found by the improved -expansion and generalized -expansion methods. By He's semi-inverse variational principle we have obtained dark and bright soliton wave solutions. Also, the obtained semi-inverse variational principle has profound implications in physical understandings. These solutions might play important role in engineering and physics fields. Moreover, by using Matlab, some graphical simulations were done to see the behavior of these solutions.