The aim of this paper is to investigate a class of generalised Kadomtsev–Petviashvili (KP) and B-type Kadomtsev–Petviashvili (BKP) equations, which include many important nonlinear evolution equations as its special cases.By applying the fundamental Pfaffian identity, a general Pfaffian formulation is established and all the involved generating functions for Pfaffian entries need to satisfy a system of combined linear partial differential equations. The illustrative examples of the presented Pfaffian solutions are given for the (3+1)-dimensional generalised KP, Jimbo–Miwa and BKP equations. Moreover, we use the linear superposition principle to generate exponential travelling wave solutions and mixed resonant solutions of the considered equations.

The aim of this paper was to study a generalised (2+1)-dimensional nonlinear Schrödinger equation, which can give us many important mathematical and physical models to illustrate various nonlinear phenomena in physical and engineering sciences. Based on a general Hirota bilinear form, the breather-type and non-singular rational solutions are presented. Akhmediev breather and Ma breather solutions can be considered as solutions representing the nonlinear propagation of the unstable model. In addition, by means of the bilinear transformation method, the fundamental rogue waves are given in terms of the Grammian determinant, which are shown to be line rouge waves. It is then demonstrated that the non-singular rational solutions generated via the long wave limit approach cover the rogue waves presented by the bilinear transformation method. The results presented in this paper exhibit the complexity and diversity of dynamical behaviour for the considered complex system.