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The dual Ito equation can be seen as a two-component generalization of the well-known Camassa-Holm equation. By using the theory of planar dynamical system, we study the existence of its traveling wave solutions. We find that the dual Ito equation has smooth solitary wave solutions, smooth periodic wave solutions, and periodic cusp solutions. Parameter conditions are given to guarantee the existence.

Study on two-component equations has drawn a lot of interest among researchers [

The Ito equation [

By the sense of “tri-Hamiltonian duality” [

Different methods have been used to study exact solutions to standard Ito equation or generalized ones as well as higher-order Ito equation [

In this paper, we will apply the method of dynamical system [

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

In this section, the properties of equilibrium points and possible phase portraits will be given.

We consider the traveling wave solutions of (

Substituting (

Integrating (

From the second equation of (

Plugging (

Letting

System (

System (

Now we consider the equilibrium points of system (

We need to find the bifurcation conditions for the parameters. Equation

Next we will study possible order of those roots. We see that

Now consider singular equilibrium points on the singular lines. On the singular line

To investigate the equilibrium points of (

By the theory of planar dynamical systems [

Except for the straight line

Let

System (

When

Phase portraits of (

When

If

If

When taking minus sign, there are two singular lines of (

When

Phase portraits of (

When

When

If

If

In this section we will give some types of interesting solutions to (

Suppose that

By using the results of the above lemmas and the basic theory of the singular nonlinear traveling wave equations [

There exists a smooth bell-shape solitary wave solution of the first component of (

Those solitary wave solutions are corresponding to the homoclinic orbit given by

Smooth bell-shape solitary wave solution.

Planar profiles of solutions to

Planar profiles of solutions to

There exists a smooth valley-shape solitary wave solution of the first component of (

These smooth valley-shape solitary wave solutions are corresponding to the homoclinic orbit given by

Smooth valley-shape solitary wave solution.

Planar profiles of solutions to

Planar profiles of solutions to

There is a family of smooth periodic wave solutions of (

Those periodic traveling wave solutions correspond to the family of smooth periodic orbits surrounding the centers in Figures

Smooth periodic wave solutions.

Planar profiles of solutions to

Planar profiles of solutions to

In Figures

There is a peaked periodic cusp wave solution if one takes minus sign in (

These peaked periodic cusp wave solutions are corresponding to the arch curve in the left side of

Peaked periodic cusp wave solution.

Planar profiles of solutions to

Planar profiles of solutions to

There is a valley-shape periodic cusp wave solution if one takes minus sign in (

Those valley-shape periodic cusp wave solutions are corresponding to the arch curve in the right side of

Valley-shape periodic cusp wave solution.

Planar profiles of solutions to

Planar profiles of solutions to

By using theory of the singular nonlinear traveling wave equations, we found the existence of several different kinds of traveling wave solutions of (

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

Research was supported by the National Natural Science Foundation of China (no. 11026169).