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By using the method of dynamical system, the exact travelling wave solutions of the coupled nonlinear Schrödinger-Boussinesq equations are studied. Based on this method, the bounded exact travelling wave solutions are obtained which contain solitary wave solutions and periodic travelling wave solutions. The solitary wave solutions and periodic travelling wave solutions are expressed by the hyperbolic functions and the Jacobian elliptic functions, respectively. The results show that the presented findings improve the related previous conclusions. Furthermore, the numerical simulations of the solitary wave solutions and the periodic travelling wave solutions are given to show the correctness of our results.

In laser and plasma physics, the significant problems under interactions between a nonlinear real Boussinesq field and a nonlinear complex Schrödinger field have been raised [

In this paper, we consider the following coupled nonlinear Schrödinger-Boussinesq equations [

The rest of this paper is built up as follows. In Section

In this section, we describe the dynamical system method for finding travelling wave solutions of nonlinear wave equations. Suppose an

Making a transformation

If system (

After

From the above description of the “three-step method,” we can see that solutions of (

Following the procedure described in Section

Use the transformation

We divided by (

Let the integral constant be

The bifurcations of phase portraits of (

From Figure

when

if

if

In the first image of Figure

We can get the following relationship from (

By using the above results and considering condition (

The 3D graphics of

The 3D graphics of

The 3D graphics of

The 3D graphics of

The 3D graphics of

The 3D graphics of

From Figures

To summarize, by using the dynamical system method, the bounded exact travelling wave solutions (solitary wave solutions and periodic wave solutions) have been obtained for the coupled nonlinear Schrödinger-Boussinesq equations. The dynamical system method is a good method to obtain exact solutions, which can not only obtain exact solutions but also understand nonlinear dynamics of travelling wave equations. We show that the hyperbolic function solutions and the Jacobian elliptic function solutions we found in this paper are different from the solutions presented by other authors before. The results enrich the diversity of wave structures of the coupled nonlinear Schrödinger-Boussinesq equations.

Furthermore, if there is not the condition that

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

The authors gratefully acknowledge the support of the National Science Foundation of China under Grant no. 11261049 and the National Science Foundation of China under Grant no. 41161001.