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The generalized HD type equation is studied by using the bifurcation method of dynamical systems. From a dynamic point of view, the existence of different kinds of traveling waves which include periodic loop soliton, periodic cusp wave, smooth periodic wave, loop soliton, cuspon, smooth solitary wave, and kink-like wave is proved and the sufficient conditions to guarantee the existence of the above solutions in different regions of the parametric space are given. Also, all possible exact parametric representations of the bounded waves are presented and their relations are stated.

The search for the exact solutions of nonlinear partial differential equations has been one of the most important concerns of mathematicians throughout the world for a long time. Because for understanding the nonlinear phenomena, which are usually described by partial differential equations, the study of the exact solutions is essential. Many powerful methods have been presented for finding the traveling wave solutions of nonlinear partial differential equations, such as the Bäcklund transformation [

The Harry Dym (HD) equation [

In 2015, Geng et al. [

Implicit soliton for

In this paper, by employing the bifurcation method of dynamical systems [

Using a transformation

Letting

From (

Using

For a fixed

Write that

Let

For an equilibrium point

Notice that for

Since both (

Bifurcation sets and phase portraits of (

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

Let

When

When

When

Taking

Periodic loop soliton of (

When

When

Taking

Periodic cusp wave of (

When

When

Taking

Smooth periodic wave of (

When

When

Taking

Loop soliton of (

When

When

Taking

Cuspon of (

When

When

When

Taking

Smooth solitary wave of (

When

When

When

When

When

When

Taking

Kink-like waves of (

For given

The level curves defined by

For given

For given

For given

For given

For given

For given

For given

For given

For given

For given

For given

Substituting (

For given

For given

Substituting (

From Figure

For given

From Figure

For given

From Figure

For given

The derivations for our main results are completed.

From Figures

From Figures

From Figure

From Figure

From Figure

From Figure

From Figure

From Figures

From Figure

Geng et al. [

Let

From (

In this paper, we investigate the dynamic properties of the generalized HD type equation (

The authors declare that there are no competing interests regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China under Grant no. 11461022 and the Natural Science Foundations of Yunnan Province, China, under nos. 2014FA037 and 2013FZ117.