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.