New exact solutions and conservation laws of a coupled Kadomtsev–Petviashvili system Abdullahi Rashid Adem, Chaudry Masood Khalique ⇑ International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa article info Article history: Received 10 October 2012 Accepted 9 April 2013 Available online 20 April 2013 Keywords: Coupled Kadomtsev–Petviashvili system Lie symmetry method ðG 0 =GÞ-expansion method Travelling waves Conservation laws Multiplier method abstract This paper obtains exact solutions of a new coupled Kadomtsev–Petviashvili system, which arises in the analysis of various problems in fluid mechanics, theoretical physics and many scientific applications. Lie symmetry method along with the ðG 0 =GÞ-expansion method is employed to find the travelling wave solu- tions of the underlying system. In addition, we derive the conservation laws of the coupled Kadomtsev– Petviashvili system using the multiplier method. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction Nonlinear partial differential equations (NLPDEs) describe a variety of physical phenomena in the fields such as physics, chem- istry, biology, and fluid dynamics. Thus finding solutions of such NLPDEs is inevitable. However, determining solutions of NLPDEs is a very difficult task and only in certain cases one can obtain exact solutions. In recent years a number of methods have been pro- posed for obtaining exact solutions of NLPDEs. Some of the most important methods found in the literature include the inverse scat- tering transform, Hirota’s bilinear method, the homogeneous bal- ance method, the Bäcklund transformation, the Darboux transformation method, the Fan sub-equation method, the new test function, the ðG 0 =GÞ-expansion method, Lie symmetry analysis, etc. [1–12]. Among the methods mentioned above, the Lie symmetry meth- od, also called the Lie group analysis method, is one of the most versatile methods to determine solutions of NLPDEs. It is based upon the study of the invariance under one parameter Lie group of point transformations [10–12]. Developed by Sophus Lie in the latter half of the nineteenth century, this method systematically unifies well known ad hoc techniques to construct explicit solu- tions for differential equations. In the last six decades there have been considerable developments in Lie symmetry methods for differential equations as can be seen by the number of research papers, books and new symbolic softwares [13–19] devoted to the subject. The celebrated Korteweg-de Vries (KdV) equation [20] u t þ 6uu x þ u xxx ¼ 0; governs the dynamics of solitary waves. It was derived to describe shallow water waves of long wavelength and small amplitude. It is an important equation from the view point of integrable systems because it has infinite number of conservation laws, gives multiple- soliton solutions, has bi-Hamiltonian structures, a Lax pair, and has many other physical properties [21]. The Kadomtsev–Petviashvili (KP) equation [22] extends the KdV equation and is given by u t þ 6uu x þ u xxx ð Þ x þ u yy ¼ 0: The KP equation is a model for shallow long waves in the x- direction with some mild dispersion in the y-direction. It is com- pletely integrable by the inverse scattering transform method and gives multiple-soliton solutions. In [23] a new coupled KdV system u t þ u xxx þ 3uu x þ 3ww x ¼ 0; ð1aÞ v t þ v xxx þ 3vv x þ 3ww x ¼ 0; ð1bÞ w t þ w xxx þ 3 2 ðuwÞ x þ 3 2 ðv wÞ x ¼ 0; ð1cÞ was formally derived and examined by using the generalized bi-Hamiltonian structures with the aid of the trace identity. In this 0045-7930/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compfluid.2013.04.005 ⇑ Corresponding author. E-mail addresses: Abdullahi.R.Adem@gmail.com (A.R. Adem), Masood.Khali- que@nwu.ac.za (C.M. Khalique). Computers & Fluids 81 (2013) 10–16 Contents lists available at SciVerse ScienceDirect Computers & Fluids journal homepage: www.elsevier.com/locate/compfluid