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 [1] -[4] to study the traveling wave solutions of the ZK-BBM equation. The generalised form of the (2 + 1) dimensional ZK-BBM equation is given as:

where and are arbitrary constants.

Equation (1) arises as a description of gravity water waves in the long-wave regime [5] [6] . The solutions of Equation (1) have been studied in various aspects. For example, Sadaf Bibi [7] used the Sine-cosine method to

obtain the travelling wave solutions of Equation (1). Rajesh Kumar Gupta [8] used the -expansion method

to find some hyperbolic, trigonometric and rational solutions, and so on. It is worth mentioning that Wazwaz [9] [10] made a detailed study for compact and noncompact physical structures and calculated the exact solutions of compact and noncompact structures by the extended tanh method for the ZK-BBM equation.

Taking the traveling wave transformation and, we can obtain the corresponding

reduced ODE of Equation (1).

Integrating Equation (2) with respect to once, we yield

where is an integral constant.

Equation (3) can be written as

where.

From Equation (4) we have

where is an integral constant.

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

Let

Then Equation (5) becomes

where is a function of. The integral form of Equation (7) is

Denote

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

Case 1. has a double real root and a simple real root. Then we have

when, the corresponding solutions are

Case 2. has a triple root. Then we have

The corresponding solution is

Case 3. has three different real roots. Then we have

when, we take the transformation as follows

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 we take the transformation as follows

The corresponding solutions is

where.

Case 4. has only a real root. Then we have

when, we take the transformation as follows

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 has been rewritten, but we still use it. The solutions are all possible exact traveling wave solutions to Equation (1). We can see it is easy to write the corresponding solutions to the ZK-BBM equation.

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.