Hybrid solutions and nonlinear transformed waves for the (3+1)-dimensional Kadomtsev–Petviashvili–Sawada–Kotera–Ramani equation

The research object of this paper is the [Formula: see text]-dimensional Kadomtsev–Petviashvili–Sawada–Kotera–Ramani (KPSKR) equation. Quite different with the other research works, the primary goal of this research is to search the hybrid solutions containing nonlinear transformed waves, lump solutions and breather solutions of the [Formula: see text]-dimensional KPSKR equation. Under premise of obtaining the N-soliton solutions which are acquired via the Hirota bilinear method, we derive lump solutions via using the long wave limit method. Also, breather solutions are attained by applying the complex conjugate construction method. Additionally, the mixed solutions containing them are derived. What’s more, six different types of nonlinear transformed waves including W-shaped soliton, quasi-anti-dark soliton, quasi-periodic soliton, multi-peak soliton, M-shaped soliton, oscillation M-shaped soliton are derived from breathers and lumps under transformation mechanism. We discuss at length the hybrid solutions containing nonlinear transformed waves when [Formula: see text]. Besides, these obtained solutions are depicted through 3D graphics, density plots and cross-sectional views.