This paper is contributed to explore all possible single peak solutions for the

It is well known that the study of nonlinear wave equations and their solutions is of great importance in many areas of physics.

In 1993, Cooper et al. [

In the development of soliton theory, there exist many different approaches to searching for exact solutions of nonlinear partial differential equations, such as mapping method [

In the literature [

In this section, we first introduce some notations. Let

Let us consider the traveling wave solution of

A function

A wave function

A wave function

Without loss of generality, one assumes that

Equation (

(i) If

If

(ii) If

If

(iii) If

Suppose that

If

If

For

For

Suppose that

If

If

If

If

or

(i) From the process of proving Theorem

(ii) If

(iii) If

(iv) If

Let

Theorem

In this case, according to Theorem

Thus we obtain the implicit solution

In view of

The phase portraits of (

The profiles of waves for

In this case, by the standard phase portrait analysis (see Figure

By virtue of Theorem

When

When

From the standard phase portrait analysis (see Figures

(1)

(2)

where

From

Corresponding to the homoclinic orbit to the saddle point

(3)

where

From

has the inverse denoted by

The profile of cuspon soliton solution is shown in Figure

(4)

(5)

From

has the inverse denoted by

The profile of cuspon soliton solution is shown in Figure

(6)

In view of

has the inverse which is denoted by

The profile of cuspon soliton solution is shown in Figure

(7)

where

In view of

has the inverse which is denoted by

The profile of smooth soliton solution is shown in Figure

Let us summarize our results in the following theorem.

Suppose that

(i)

(ii)

(iii)

with the following properties:

with the following properties:

with the following properties:

with the following properties:

with the following properties:

with the following properties:

In this paper, we study the single peak solitary wave solutions of

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China (11361017, 11161013), the Natural Science Foundation of Guangxi (2012GXNSFAA053003, 2013GXNSFAA019010), and the Innovation Project of GUET Graduate Education (XJYC2012021, XJYC2012022).