Three-wave resonant interactions: dark–bright–bright mixed N- and high-order solitons, breathers, and their structures

Abstract

In this paper, a three-wave resonant interaction system, which describes the mixture of waves with different frequencies in a weakly nonlinear and dispersive medium, is investigated with the help of the Darboux transformation and its extension. Firstly, we construct the dark–bright–bright mixed multi-soliton solutions in terms of the determinant. Dark–bright–bright solitons are found to arise in the focusing, defocusing and mixed cases. Bound state of the two dark–bright–bright solitons is also analyzed. Secondly, we derive the dark–bright–bright mixed high-order semi-rational solitons and find that their structures are different from the multi-soliton ones. Finally, we present the Nth-order breather solutions in terms of the determinant, where N is a positive integer. Line-shaped and three wing-shaped structures of the breathers are obtained.

Keywords:

• Three-wave resonant interactions
• dark–bright–bright mixed multi-solitons
• dark–bright–bright mixed high-order solitons
• three wing-shaped structures of the breathers

Acknowledgments

We express our sincere thanks to all the members of our discussion group for their valuable comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work has been supported by the National Natural Science Foundation of China [grant number 11772017] and by the Fundamental Research Funds for the Central Universities.