Exploring soliton dynamics and wave interactions in an extended Kadomtsev-Petviashvili-Boussinesq equation

This research investigates the extended Kadomtsev-Petviashvili-Boussinesq equation, relevant in numerous scenarios involving dissipative media. To initiate the analysis, a Hirota bilinear form is applied, leading to a Bäcklund transformation for the equation. Utilizing this bilinear Bäcklund transformation, we derive solutions comprising rational and exponential functions. Moreover, the Hirota bilinear representation is employed extensively to decipher the dynamics of two-wave, three-wave, rogue, lump-kink, breather, and multi-wave solutions. Detailed visualizations, including contour plots and 3D profiles across diverse parameter settings, offer deeper insights into the characteristics of these results. A traveling wave transformation further simplifies the equation into an ordinary differential equation, and the use of a new auxiliary equation method yields solutions that encompass trigonometric, hyperbolic, rational, and exponential functions. Our results highlight the effectiveness and efficiency of these methods in analytically solving nonlinear partial differential equations.