^{1}

^{2}

^{1}

^{2}

We consider a modification of the

To study the role of nonlinear dispersion in the formation of patterns in the liquid drop, Rosenau and Hyman [

In this paper, we consider the following modification of the

Let

Integrating (

Letting

System (

We know that a peakon solution of (

The rest of this paper is organized as follows. In Section

Using the transformation

when

when

when

and when

Next, we focus on the cases

Write

Let

By the theory of planar dynamical systems, we know that, for an equilibrium point

For a fixed

Using the property of equilibrium points and bifurcation theory, we obtain the following results.

When

When

The bifurcation sets and phase portraits of system (

The bifurcation sets and phase portraits of system (

The bifurcation sets and phase portrait of system (

In this section, we present some exact peakon, solitary, and smooth periodic wave solutions of (

(i) For given

The level curves of system (

Substituting (

Completing the above integral and solving the equation for

Noting that

Peakons, solitary, and smooth periodic waves of (

(ii) For given

Substituting (

Completing the above integral, we can get a peakon solution as follows:

(iii) For given

Substituting (

Completing the above integral, we can get a peakon solution as follows:

(iv) For given

Substituting (

Completing the above integral, we can get a peakon solution as follows:

(i) For given

Substituting (

Completing the above integral, we can get a solitary wave solution as follows:

(ii) For given

Substituting (

Completing the above integral, we can get a solitary wave solution as follows:

(iii) For given

Substituting (

Completing the above integral, we can get a solitary wave solution as follows:

(iv) For given

Substituting (

Completing the above integral, we can get a solitary wave solution as follows:

(v) For given

Substituting (

Completing the above integral, we can get a solitary wave solution as follows:

(vi) For given

Substituting (

Completing the above integral, we can get a solitary wave solution as follows:

From Figures

(i) For given

Substituting (

Completing the above integral, we can get a peakon solution as follows:

Substituting (

Completing the above integral, we can get a solitary wave solution as follows:

(ii) For given

Substituting (

Completing the above integral, we can get a peakon solution as follows:

Substituting (

Completing the above integral, we can get a solitary wave solution as follows:

(i) For given

Substituting (

Completing the above integral, we can get a smooth periodic wave solution as follows:

(ii) For given

Substituting (

Completing the above integral, we can get a smooth periodic wave solution as follows:

(iii) For given

Substituting (

Completing the above integral, we can get a smooth periodic wave solution as follows:

(iv) For given

Substituting (

Completing the above integral, we can get a smooth periodic wave solution as follows:

In this paper, using the bifurcation theory and the method of phase portraits analysis, we investigated a modification of the

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

The authors would like to thank the reviewers very much for their useful comments and helpful suggestions. This work is supported by the Natural Science Foundation of Yunnan Province, China (no. 2013FZ117), and the National Natural Science Foundation of China (no. 11161020).