One-dimensional three-wave resonant interaction systems: General higher-order semi-rational rogue wave solutions and their dynamic behaviors

Source abstract

General higher-order semi-rational rogue wave solutions within one-dimensional three-wave resonant interaction systems are investigated, utilizing 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) N1-th order rogue waves with one simple root, accompanied by N2-th order breathers and N3-th order dark solitons as coexisting backgrounds; (ii) (Ñ1,Ñ2)-th order rogue waves with two simple roots, accompanied by N2-th order breathers as backgrounds; and (iii) (Ñ1,Ñ2)-th order rogue waves with one double root, accompanied by N2-th order breathers as backgrounds. The results demonstrate that pure rogue waves, pure breathers, pure dark solitons, and their various combinations can be derived from these general higher-order semi-rational rogue wave solutions by selecting appropriate nonzero integers N1, N2, N3, Ñ1 and Ñ2. A significant advancement is the introduction of generalized tau functions, which improve the representation of diverse solitary wave solutions within the Kadomtsev-Petviashvili reduction framework. This methodological innovation applies to other one-dimensional integrable systems, expanding the exploration of general higher-order semi-rational rogue wave solutions. As an application, the intricate patterns and dynamic behaviors of these rogue waves, accompanied by dark solitons, breathers, and their coexistence, are analyzed and visualized through several illustrative figures.