This paper investigates general higher-order semi-rational solutions to the (2+1)-dimensional three-wave resonant interaction systems, employing the bilinear Kadomtsev–Petviashvili hierarchy reduction method. These solutions, formulated as determinants involving Schur polynomials and exponential functions, are classified into three distinct categories: (i) \(N_{1}\) -th order line rogue waves (or lumps) with \(N_{3}\) -th order dark solitons as backgrounds; (ii) \(N_{1}\) -th order line rogue waves (or lumps) with \(N_{2}\) -th order breathers as backgrounds; and (iii) \(N_{1}\) -th order line rogue waves (or lumps) with both \(N_{2}\) -th order breathers and \(N_{3}\) -th order dark solitons as coexisting backgrounds. By selecting appropriate nonzero integers \(N_{1}\) , \(N_{2}\) , and \(N_{3}\) , pure line rogue waves, pure lumps, pure breathers, pure dark solitons, and their various combinations can be derived from these general higher-order semi-rational solutions. As an application, the intricate patterns and dynamic behaviors of these line rogue waves (or lumps), with backgrounds consisting of breathers, dark solitons, and their combinations, are analyzed and visualized through several illustrative figures.
