Symmetry reductions, exact equations and the conservation laws of the generalized (3+1) dimensional Zakharov-Kuznetsov equation

Abstract

Because the nonlinear evolution equations can describe the complex phenomena of physical, chemical and biological field, many methods have been proposed for investigating such types of equations, and the Lie symmetry analysis method is one of the powerful tools for studying the nonlinear evolution equations. By using the Lie symmetry analysis method, we can obtain the symmetries, reduced equations, group invariant solutions, conservation laws, etc. In the reduction process, we can reduce the order and dimension of the equations, and a complex partial differential equations (PDE) can be reduced to ordinary differential equations directly, which simplifies the solving process. Meanwhile, the symmetries, conservation laws and exact solutions to the nonlinear partial differential equations play a significant role in nonlinear science and mathematical physics. For example, we can obtain a lot of new exact solutions by the known symmetries of the original equation; through the analysis of the special form of solution we can better explain some physical phenomena. In addition, the studying of conservation laws and symmetry groups is also the central topic of physical sciencein both classical mechanics and quantum mechanics. Lie symmetry analysis method is suitable for not only constant coefficient equations, but also variable coefficient equations and PDE systems. By using Lie symmetry analysis method, the symmetries and corresponding symmetry reductions of the (3+1) dimensional generalized Zakharov-Kuzetsov (ZK) equation are obtained. Combining the homogeneous balance principle, the trial function method and exponential function method, the group invariant solutions and some new exact explicit solutions are obtained, including the shock wave solutions, solitary wave solutions, etc. Then, we give the conservation laws of the generalized (3+1) dimensional ZK equation in terms of the Lagrangian and adjoint equation method.