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The purpose of this paper is to explore new reduced form of the (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili (BKP) equation by considering its bilinear equations, derived from connection between the Hirota’s transformation and Bell polynomials.

Based on the bilinear form of new reduced form of the (3 + 1)-dimensional generalized BKP equation, lump solutions with sufficient and necessary conditions to guarantee analyticity and rational localization of the solutions are discovered. Also, extended homoclinic approach is applied to considered equation for finding solitary wave solutions.

A class of the bright-dark lump waves are fabricated for studying different attributes of (3 + 1)-dimensional generalized BKP equation and some new exact solutions including kinky periodic solitary wave solutions and line breathers periodic are also obtained by Following the extended homoclinic approach.

The paper presents that the implemented methods have emerged as a promising and robust mathematical tool to manage (3 + 1)-dimensional generalized BKP equation by using the Hirota’s bilinear equation.

By considering important characteristics of lump and solitary wave solutions, one can understand the shapes, amplitudes and velocities of solitons after the collision with another soliton.

The analysis of these higher-dimensional nonlinear wave equations is not only of fundamental interest but also has important practical implications in many areas of mathematical physics and ocean engineering.

To the best of the authors’ knowledge, the acquired solutions given in various cases have not been reported for new reduced form of the (3 + 1)-dimensional generalized BKP equation in the literature. These obtained solutions are advantageous for researchers to know objective laws and grab the indispensable features of the development of the mathematical physics.

This paper aims to study the breather, lump-kink and interaction solutions of a (3 + 1)-dimensional generalized shallow water waves (GSWW) equation, which describes water waves propagating in the ocean or is used for simulating weather.

Hirota bilinear form and the direct method are used to construct breather and lump-kink solutions of the GSWW equation. The “rational-cosh-cos-type” test function is applied to obtain three kinds of interaction solutions.

The fusion and fission of the interaction solutions between a lump wave and a 1-kink soliton of the GSWW equation are studied. The dynamics of three kinds of interaction solutions between lump, kink and periodic waves are discussed graphically.

This paper studies the breather, lump-kink and interaction solutions of the GSWW equation by using various approaches and provides some phenomena that have not been studied.

The purpose of this study is to examine the lump solutions of the (3 + 1)-dimensional nonlinear evolution equations by considering a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation and a (3 + 1)-dimensional variable-coefficient generalized B-type Kadomtsev–Petviashvili (vcgBKP) equation as examples.

Based on Hirota’s bilinear theory, a direct method is used to examine the lump solutions of these two equations.

The complete non-elastic interaction solutions between a lump and a stripe are also discussed for the equations, which show that the lump solitons are swallowed by the stripe solitons.

The dynamics of these solutions are analyzed to enrich the diversity of the dynamics of high-dimensional KP-type nonlinear wave equations.