Bilinear form, soliton, breather, lump and hybrid solutions for a ($$\varvec{2+1}$$)-dimensional Sawada–Kotera equation

In this paper, we investigate a ( $$2+1$$ )-dimensional Sawada–Kotera (SK) equation for the atmosphere, rivers, lakes, oceans, as well as the conformal field and two-dimensional quantum gravity gauge field. Bilinear form and N-soliton solutions, which are different from those in the existing literatures, are derived, where N is a positive integer. The higher-order breather, lump and hybrid solutions for the ( $$2+1$$ )-dimensional SK equation are also constructed based on the N-soliton solutions. Three kinds of the first-order breathers are obtained, and the higher-order breathers are constructed. The higher-order lump solutions are also derived via the long-wave limit method. Hybrid solutions composed of the solitons, breathers and lumps are worked out, and interaction between the waves is discussed graphically. Finally, similar solutions for a generalized form of the ( $$2+1$$ )-dimensional SK equation are given.