Exact Travelling Wave Solutions for a Modified Zakharov–Kuznetsov Equation

The modied Zakharov–Kuznetsov (mZK) equation, ut + uux + uxxx + uxyy = 0, (1) represents an anisotropic two-dimensional generalization of the Korteweg–de Vries equation and can be derived in a magnetized plasma for small amplitude Alfvén waves at a critical angle to the undisturbed magnetic field, and has been studied by many authors because of its importance [1–5]. However, Eq. (1) possesses many interesting traveling wave structures which have not yet been found. In the study of equations modeling wave phenomena, one of the fundamental objects of study is the traveling wave solution, meaning a solution of constant form moving with a fixed velocity. Traveling waves, whether their solution expressions are in explicit or implicit forms, are very interesting from the point of view of applications. These types of waves will not change their shapes during propagation and are thus easy to detect. Of particular interest are three types of traveling waves: the solitary waves, which are localized traveling waves, asymptotically zero at large distances, the periodic waves, and the kink waves, which rise or descend from one asymptotic state to another. Recently, a unified method, called the extended mapping method, is developed to obtain exact traveling wave solutions for a large variety of nonlinear partial differential equations [6–8]. By means of this method, the solitary wave, the periodic wave and the kink wave (or the shock wave) solutions can, if they exist, be obtained simultaneously to the equation in question. Thus, many of tedious and repetitive calculations may be avoided. The method is further developed in this paper to study the traveling wave solution of Eq. (1). For a given nonlinear evolution equation, say, in three independent variables