In this paper, we apply Hirota’s bilinear method in conjunction with the Kadomtsev–Petviashvili hierarchy reduction technique to construct solitons, breathers and lump solutions of the coupled (2+1)-dimensional Fokas system. These solutions are expressed in \(N\times N\) Gram-type determinants. Dynamics of the one and two solitons, breathers and lumps are investigated. In addition, we derive three types of breather solutions by different choices of parameters, including Akhmediev breather, Kuznetsov-Ma breather and general one along some oblique line. By introducing two judicious differential operators in the dimension reduction procedure, we obtain rational solutions in terms of Schur polynomials. Three different kinds of fundamental lumps including bright lumps, dark lumps and bi-model lumps are investigated. The superposition of fundamental lumps generate higher-order lump solutions, and we show that the arrangement patterns of fundamental lumps are determined by the irreducible parameters. Pattern formation of higher-order lump solutions at large times is also studied, which is described analytically by root structures of the Yablonskii–Vorob’ev polynomials. Finally, we propose a multi-component Fokas system and present multi-soliton and multi-breather solutions.
