Dynamics of solitons, multi-lumps, and their interactions of the Konopelchenko-Dubrovsky-Kaup-Kupershmidt and Bogoyavlensky-Konopelchenko model in (3+1)-dimensions using the modern advanced approach

In this original research, we explore new forms of soliton solutions, lumps, and the dynamics of water waves of the considered Generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt (KDKK) and Bogoyavlensky-Konopelchenko (BK) equations in (3+1)-dimensions, utilizing the Hirota bilinear approach. We apply the Painlevé analysis to check the integrability of this model. We formulate a bilinear equation in an auxiliary function using the Cole-Hopf transformation and then develop it into a Hirota bilinear form using a bilinear differential operator. Based on this technique, we derive lump waves, breaking phenomena, solitons, peakons, wave interactions, and wave-to-wave collisions. The solutions are obtained using ansatz functions in quadratic, sine, cosine, and exponential functions. We also illustrate how the lump interacts with solitary, periodic, and breather waves to generate various dynamics of water waves in the obtained solutions, which are visualized graphically via 2D, 3D, and contour plots with the help of symbolic software Maple. We also show the phase plane portraits, bifurcation analysis, and sensitivity analysis based on the bifurcation method with the help of the equilibrium points of the governing model studied.