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In our work, a generalized KdV type equation of neglecting the highest-order infinitesimal term, which is an important water wave model, is discussed by using the simplest equation method and its variants. The solutions obtained are general solutions which are in the form of hyperbolic, trigonometric, and rational functions. These methods are more effective and simple than other methods and a number of solutions can be obtained at the same time.

In recent decades, the study of nonlinear partial differential equations (NLEEs) modelling physical phenomena has become an important research topic. Seeking exact solutions of NLEEs has long been one of the central themes of perpetual interest in mathematics and physics. With the development of symbolic computation packages like Maple and Mathematica, many powerful methods for finding exact solutions have been proposed, such as the homogeneous balance method [

The simplest equation method is a very powerful mathematical technique for finding exact solutions of nonlinear ordinary differential equations. It has been developed by Kudryashov [

Recently, Bilige et al. introduced a method called the extended simplest equation method, as an extension of the simplest equation method, to look for the exact traveling wave solutions of NLEEs [

In 1995, based on the physical and asymptotic considerations, Fokas [

Assuming that the waves are unidirectional and neglecting terms of

Assuming that the

Equation (

If only we neglect the highest-order term of

In fact, (

Of course, on describing dynamical behaviors of water waves, (

Equation (

The organization of the paper is as follows. In Section

Consider a general nonlinear partial differential equation (PDE) for

By taking

Suppose the solution

The Bernoulli equation we consider in this paper is

For the Riccati equation

Substitute (

Assuming that the constants

In (

Equation (

We Suppose the solution

To determine the positive integer

Suppose that the value of the constants

The general solutions of (

when

when

when

We Suppose the solution

By substituting (

Assume constants

Making a transformation

Considering the homogeneous balance between

Substituting (

On solving the above algebraic equations using the Maple, we get the following results:

Therefore, using solutions (

Suppose the solution of (

Substituting (

On solving the above algebraic equations using the Maple, we get the following results:

Therefore, using solutions (

Suppose the solution of (

Substituting (

Substituting (

Substituting (

Substituting (

Suppose the solution of (

By substituting (

If

Substituting (

If

Substituting (

If

Substituting (

In this section, some typical wave figures are given as in Figures

2D and 3D figures of solution

2D and 3D figures of solution

3D figures of solution

2D figures of solution

2D figures of solution

In this paper, with the aid of Maple, we successfully obtained wider classes of exact traveling solutions of (

The author declares that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (11161020; 11361023), the Natural Science Foundation of Yunnan Province (2011FZ193), and the Natural Science Foundation of Education Committee of Yunnan Province (2012Y452 and 2013C079).