The Klein–Gordon–Zakharov equations with the positive fractional power terms and their exact solutions

In this paper, the famous Klein–Gordon–Zakharov (KGZ) equations are first generalized, and the new special types of KGZ equations with the positive fractional power terms (gKGZE) are presented. In order to derive exact solutions of the new special gKGZE, subsidiary higher-order ordinary differential equations (sub-ODEs) with the positive fractional power terms are introduced, and with the aid of the sub-ODE, exact solutions of four special types of the gKGZE are derived, which are the bell-type solitary wave solution, the algebraic solitary wave solution, the kink-type solitary wave solution and the sinusoidal travelling wave solution, provided that the coefficients of gKGZE satisfy certain constraint conditions.