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In order to get the traveling wave solutions of the Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZK-BBM) equation, it is reduced to an ordinary differential equation (ODE) under the travelling wave transformation first. Then complete discrimination system for polynomial is applied to the ZK-BBM equation. The traveling wave solutions of the equation can be obtained.

The nonlinear partial differential equation (PDE) is widely used to describe physical phenomena in various fields of sciences, especially in fluid mechanics, solid state physics, plasma physics, plasma waves, biology and so on. During the past few decades, various methods have been developed by researchers to find the solutions for the NLEEs.

In this article, we will use complete discrimination system for polynomial proposed by Liu [

where

Equation (1) arises as a description of gravity water waves in the long-wave regime [

obtain the travelling wave solutions of Equation (1). Rajesh Kumar Gupta [

to find some hyperbolic, trigonometric and rational solutions, and so on. It is worth mentioning that Wazwaz [

Taking the traveling wave transformation

reduced ODE of Equation (1).

Integrating Equation (2) with respect to

where

Equation (3) can be written as

where

From Equation (4) we have

where

We use the complete discrimination system for the third order polynomial and have the following solving process.

Let

Then Equation (5) becomes

where

Denote

According to the complete discrimination system, we give the corresponding single traveling wave solutions to Equation (1).

Case 1.

when

Case 2.

The corresponding solution is

Case 3.

when

According to the Equation (8), we have

where

On the basis of Equation (19) and the definition of the Jacobi elliptic sine function, we have

The corresponding solution is

when

The corresponding solutions is

where

Case 4.

when

According to the Equation (8), we have

where

On the basis of Equation (26) and the definition of the Jacobi elliptic cosine function, we have

The corresponding solutions is

In Equations (12), (13), (14), (16), (21), (23) and (28), the integration constant

In this article, the traveling wave solutions to ZK-BBM equation were obtained by the complete discrimination system for polynomial and direct integral method. This method has the characteristics of simple steps and clear effectivity. In this way we can solve a lot of other equations.

I would like to thank the referees for their valuable suggestions.