We present the novel generalized perturbation (n, M  )-fold Darboux transformations (DTs) for the (2+1$2+1$)-dimensional Kadomtsev–Petviashvili (KP) equation and its extension.

The higher-order rational solitons and rogue wave solutions of the KP equation and its extension are found in terms of determinants, which display abundant interesting RW and soliton structures including the triangle, pentagon, heptagon, parallel profiles, etc.

We find that the new phenomenon that the parameter (a) can control the wave structures of the KP equation from higher-order rogue waves (a≠$\ne$0) into higher-order rational solitons (a=0$a=0$) in (x,t)-space with y = const.