Some Reduction and Exact Solutions of a Higher-Dimensional Equation

Abstract

The conservation laws of the $(3+1)$-dimensional Zakharov-Kuznetsov equation were obtained using Noether’s theorem after an interesting substitution $u=v_x$ to the equation. Then, with the aid of an obtained conservation law, the generalized double reduction theorem was applied to this equation. It can be verified that the reduced equation is a second order nonlinear ODE. Finally, some exact solutions of the Zakharov-Kuznetsov equation were constructed after solving the reduced equation.