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An integrable 2-component Camassa-Holm (2-CH) shallow water system is studied by using integral bifurcation method together with a translation-dilation transformation. Many traveling wave solutions of nonsingular type and singular type, such as solitary wave solutions, kink wave solutions, loop soliton solutions, compacton solutions, smooth periodic wave solutions, periodic kink wave solution, singular wave solution, and singular periodic wave solution are obtained. Further more, their dynamic behaviors are investigated. It is found that the waveforms of some traveling wave solutions vary with the changes of parameter, that is to say, the dynamic behavior of these waves partly depends on the relation of the amplitude of wave and the level of water.

In this paper, employing the integral bifurcation method together with a translation-dilation transformation, we will study an integrable 2-component Camassa-Holm (2-CH) shallow water system [

Although (

It is worthy to mention that the solutions obtained by us in this paper are different from others in existing references, such as [

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

Obviously, (

After substituting (

Let

Making a transformation

Obviously, systems (

In this section, we will investigate different kinds of exact traveling wave solutions for (

It is easy to know that (

We mainly aim to consider the new results of (

Under the parametric conditions

If

If

If

If

If

If

The profiles of multiwaveform of (

Antikink wave

Transmutative wave of antikink waveform

Dark soliton of thin waveform

Dark soliton of fat waveform

Compacton

Loop soliton of fat waveform

Loop soliton of thin waveform

Loop soliton of oblique waveform

The profiles of multiwaveform of (

Kink wave

Bright soliton of fat waveform

Bright soliton of thin waveform

Singular wave of cracked loop soliton

The profiles of multiwaveform of the first solution of (

Bright soliton

Bright compacton

Loop soliton of fat waveform

Loop soliton of thin waveform

The profiles of kink wave solution (the first solution of (

Smooth kink wave

Smooth kink wave

We observe that some profiles of above soliton-like solutions are very much sensitive to one of parameters, that is, their profiles are transformable (see Figures

From Figures

Similarly, from Figures

As in Figure

Different from the properties of solutions (

Under the parametric conditions

(i) If

Substituting (

(ii) If

As in the first case (i) using the same method, we obtain a couple of periodic solutions of (

(iii) If

(iv) If

(v) If

As in the first case (i) using the same method, we obtain a couple of periodic solutions of (

In order to show the dynamic properties of above periodic solutions intuitively, as examples, we plot their graphs of the solutions (

The profiles of smooth periodic waves of solutions (

Smooth periodic wave

Smooth periodic wave

The profiles of noncontinuous periodic waves of solutions (

Periodic kink wave

Singular periodic wave

Figures

From the above illustrations, we find that the waveforms of some solutions partly depend on wave parameters. Indeed, in 2006, Vakhnenko and Parkes’s work [

Evolvement graphs of 3D-curves along with the time and their projection curve.

3D-curves of solution (

3D-curves and their projection

In this work, by using the integral bifurcation method together with a translation-dilation transformation, we have obtained some new traveling wave solutions of nonsingular type and singular type of 2-component Camassa-Holm equation. These new solutions include soliton solutions, kink wave solutions, loop soliton solutions, compacton solutions, smooth periodic wave solutions, periodic kink wave solution, singular wave solution, and singular periodic wave solution. By investigating these new exact solutions of parametric type, we found some new traveling wave phenomena, that is, the waveforms of some solutions partly depend on wave parameters. For example, the waveforms of solutions (

The authors thank reviewers very much for their useful comments and helpful suggestions. This work was financially supported by the Natural Science Foundation of China (Grant no. 11161020) and was also supported by the Natural Science Foundation of Yunnan Province (no. 2011FZ193).