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Based on the Hirota bilinear form, two classes of lump-type solutions of the (4[Formula: see text]+[Formula: see text]1)-dimensional nonlinear Fokas equation, rationally localized in almost all directions in the space are obtained through a direct symbolic computation with Maple. The resulting lump-type solutions contain free parameters. To guarantee the analyticity and rational localization of the solutions, the involved parameters need to satisfy certain constraints. A few particular lump-type solutions with special choices of the involved parameters are given.

In this paper, a (3 + 1)-dimensional generalized B-type Kadomtsev–Petviashvili (BKP) equation is mainly discussed. Based on the Wronskian technique, a Wronskian formulation is established. Generating functions for matrix entries satisfy a linear system of partial differential equations involving a free parameter. The resulting solutions formulae provide us with a comprehensive approach to constructing rational solutions, positons and complexitons for the (3 + 1)-dimensional generalized BKP equation.

a b s t r a c tThe multiple Exp-function method is used to construct multiple wave solutions to the (3 + 1)-dimensional generalized BKP equation. The resulting solutions involve generic phase shifts and wave frequencies containing some existing choices. By taking the standard truncated Painlevé analysis, we obtained an auto-Bäcklund transformation and some types of exact solutions of the (3 + 1)-dimensional generalized BKP equation. Moreover, the linear superposition principles of hyperbolic and trigonometric function solutions are also presented.