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In this paper, based on a bilinear differential equation, we study the breather wave solutions by employing the extended homoclinic test method. By constructing the different forms, we also consider the interaction solutions. Furthermore, it is natural to analyse dynamic behaviors of three-dimensional plots.

Recently, great attention has been paid to the study about exact solutions of nonlinear partial differential equations. So, it becomes more important to seek exact solutions of nonlinear partial differential equations (NLPDEs), which occur in many fields, such as chemistry, biology, optics, classical mechanics, acoustics, engineering, and social sciences. At present, many mathematicians have proposed a large number of methods to seek exact solutions, such as Bäcklund transformation [

The two mixed Calogero-Bogoyavlenskii-Schiff (CBS) and Bogoyavlensky-Konopelchenko (BK) equations [

If we take

In this section, we will use the extended homoclinic text method [

Substituting Equation (

Spatiotemporal structure of solution (

Substituting Equation (

Spatiotemporal structure of solution (

Substituting Equation (

Substituting

Spatiotemporal structure of solution (

The evolution of solution (

Spatiotemporal structure of solution (

Substituting Equation (

The evolution of solution (

Spatiotemporal structure of solution (

Substituting Equation (

Spatiotemporal structure of solution (

The figure is given as Figure

Spatiotemporal structure of solution (

The figure is drawn as Figure

Spatiotemporal structure of solution (

With the help of Maple, we will discuss the interaction between a lump and one-kink soliton by taking

In order to obtain the dynamic feature, we choose Case

Spatiotemporal structure of solution (

In order to get interaction solutions between a lump and periodic waves, we will take

When we change the coefficients of the equation, the value of Equation (

With the help of Maple, the three-dimensional dynamic graphs are drawn as Figure

Spatiotemporal structure of solution (

In this paper, based on a bilinear differential equation, we study the breather wave solutions and the interaction solutions of the mixed Calogero-Bogoyavlenskii-Schiff and Bogoyavlensky-Konopelchenko equations. Compared with the existing results in the literature, our results are new. It will be extensively used to report many attractive physical phenomena in the fields of acoustics, heat transfer, fluid dynamics, classical mechanics, and so on. It is demonstrated that the Hirota operators are very simple and powerful in constructing new nonlinear differential equations, which possess nice math properties. It is interesting to study the interaction solutions between soliton solutions and period solution by making

No data were used to support this study.

The authors declare that they have no conflicts of interest.

The work is supported by the National Natural Science Foundation of China (project Nos. 11371086, 11671258, and 11975145), the Fund of Science and Technology Commission of Shanghai Municipality (project No. 13ZR1400100), the Fund of Donghua University, Institute for Nonlinear Sciences, and the Fundamental Research Funds for the Central Universities.