In this paper, the characteristic function method is applied to seek traveling wave solutions of nonlinear partial differential equations in a unified way. We consider the Wu-Zhang equation (which describes (1 + 1)-dimensional dispersive long wave). The equations governing the wave propagation consist of a pair of non linear partial differential equations. The characteristic function method reduces the system of nonlinear partial differential equations to a system of nonlinear ordinary differential equations which is solved via the shooting method, coupled with Runge-kutta scheme. The results include kink-profile solitary wave solutions, periodic wave solutions and rational solutions. As an illustrative example, the properties of some soliton solutions for Wu-Zhang equation are shown by some figures.

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This paper is devoted to the analysis of the two-layer shallow-water equations representing gravity currents. A similarity technique which is the characteristic function method is applied for this study. The application of the characteristic function method makes it possible to obtain the similarity forms depending on a group of infinitesimal transformations. Thus, the number of independent variables is reduced by one and the governing partial differential equations with the auxiliary conditions reduce to a system of ordinary differential equations with the appropriate auxiliary conditions. Numeric solutions are presented and discussed