In this paper, we look at a class of two-parameter coupled Kirchhoff-type fractional differential equations. Two differentiated methods are used to prove the existence of two solutions to the equation. The fundamental difference between the two methods is that the first provides asymptotic conditions for the non-linear terms on the right-hand side of the equation, while the second provides algebraic conditions; both methods combine substantial A-R conditions.

In this paper, the semi-rational solutions that causes vector rogue waves and breathers can be obtained by using the Darboux dressing transformation. We studied vector rogue waves and the interaction between rogue waves and light-dark solitons, and observed that during the interaction, due to the interference between the light-dark components of the solitons, a respiration-like structures appears. Besides, it can be observed that the rogue waves and soliton merge together. Moreover, the main characteristics of the interactions between the breathers and bright-dark solitons are displayed with some graphics.