ISSN 1541-308X, Physics of Wave Phenomena, 2011, Vol. 19, No. 2, pp. 148–154. c  Allerton Press, Inc., 2011. SOLITONS AND CHAOS Solutions of Zakharov−Kuznetsov Equation with Power Law Nonlinearity in (1+3) Dimensions B. T. Matebese 1 , A. R. Adem 1 , C. M. Khalique 1 , and A. Biswas 1,2* 1 International Institute for Symmetry Analysis and Mathematical Modeling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho 2735, Republic of South Africa 2 Department of Mathematical Sciences, Center for Research and Education in Optical Sciences and Applications, Delaware State University, Dover, DE 19901-2277, USA Received November 23, 2010 Abstract—This paper studies the Zakharov−Kuznetsov equation in (1+3) dimensions with an arbitrary power law nonlinearity. The method of Lie symmetry analysis is used to carry out the integration of the Zakharov−Kuznetsov equation. The solutions obtained are cnoidal waves, periodic solutions, singular periodic solutions, and solitary wave solutions. Subsequently, the extended tanh-function method and the G ′ /G method are used to integrate the Zakharov−Kuznetsov equation. Finally, the nontopological soliton solution is obtained by the aid of ansatz method. There are numerical simulations throughout the paper to support the analytical development. DOI: 10.3103/S1541308X11020117 1. INTRODUCTION The theory of nonlinear evolution equations (NLEEs) have come a long way through. This is one of the most important areas of research in applied mathematics for the past few decades. There are various kinds of such equations that are studied in this area [1−21]. One of the main issues that are studied in this context is the integration aspect of these equations. Once upon a time, there was this classic method of integration that is known as the inverse scattering transform (IST) which was used to carry out the integration of NLEEs. If an equation was integrable by this method, then one could work with a closed form solution, or else one had to stay contended with the numerical solutions. Such days of sufferings are over. Today, there are several mathe- matical tools that are developed across the whole wide world to carry out the integration of various NLEEs. These modern mathematical methods are indeed a true blessing in the area of NLEEs. It must be noted that there are several advantages of these methods over the IST technique. These techniques are utilized to calculate solutions to the NLEEs. Some of these modern methods of integra- tion are variational iteration method, collective vari- ables approach, semi-inverse variational principle, exponential function approach, sub-ODE method, * E-mail: biswas.anjan@gmail.com G ′ /G method, tanh−cotanh method, sine−cosine method, Lie symmetry analysis, Hirota’s bilinear approach, Riccati’s equation method and many more. However, one must exercise caution while using these methods as it could lead to incorrect results as pointed out by Kudryashov [10]. Additionally, these various methods of integration obtain multiple types of solutions to a NLEE which was simply not possible by the dominant method of IST. Some of these solutions are cnoidal wave so- lutions, compacton solutions, singular solitons, peri- odic solutions, rational solutions, and many more. In this paper, a few such methods of integration will be utilized to integrate one such NLEE. The NLEE to be studied in this paper is the Za- kharov−Kuznetsov (ZK) equation. With regards to the applications of the ZK equation, it governs dust acoustic solitary waves that are derived for magne- tized dusty plasmas which contains N different dust grains with a size distribution and charge fluctua- tion, nonthermally distributed ions and Boltzmann distributed electrons [21]. ZK equation also governs the behavior of weakly nonlinear ion-acoustic waves in a plasma comprising cold ions and hot isother- mal electrons in the presence of a uniform magnetic field [12, 14]. The quantum ZK equation is studied in the context of finite, but small amplitude of acous- tic waves in quantum magnetoplasmas [13]. In this paper, the ZK equation is studied in (1+3) dimen- 148