Exploring Wronskian, soliton, lump and lump-multi-kink solutions of a (\(3 + 1\))-dimensional Boussinesq type equation uisng Hirota bilinear and Riccati methods

This article investigates a \((3+1)\)-dimensional Boussinesq-type equation (BTE) that models the propagation of small-amplitude dispersive waves in shallow water. The equation is of great physical significance due to its relevance in diverse phenomena such as ocean wave dynamics, tsunami propagation, and coastal harbor behavior. By establishing the intrinsic relationship between the Bell polynomials and the Hirota D-operator, we construct the Hirota bilinear form of the equation in a systematic manner. Based on this bilinear framework, we derive and graphically illustrate various nonlinear wave structures, including one-, two-, and three-kink soliton solutions. Furthermore, a lump solution is obtained through a quadratic test function, and its strong localization characteristics are demonstrated through detailed graphical visualization. Two distinct families of lump-multi-kink interaction solutions are then derived by employing quadratic–exponential and quadratic–hyperbolic cosine test functions, respectively, revealing rich and complex wave interaction dynamics. Furthermore, the application of the Plücker relation leads to the Wronskian condition, demonstrating that the N-soliton solutions of the model can be represented through Wronskian determinants. Finally, by invoking the Riccati equation approach, we derive several new classes of soliton solutions expressed in hyperbolic, trigonometric, and rational function forms, enriching the overall solution structure of the model.