Exploration of lump, interactions and rogue wave solutions to the (2 + 1)-dimensional evolution equation with bifurcation, sensitivity and stability analysis

In this manuscript, for the intellectual curiosity, we analyze the (2+1)-dimensional evolution equation, namely the Kadomtsev-Petviashvili (KP) equation, which is a significant model in nonlinear physics, describing the dynamics of weakly nonlinear, dispersive waves. By employing the Hirota bilinear method and the modified Sardar subequation (MSSE) method, we derive lump as well as different exact solitary wave solutions of the selected model, which are not documented in previous literature. We manifested some novel lump soliton solutions, including homoclinic breather wave, two-wave solution, M-shaped solutions such as the M-shaped rational solution, M-shaped rational solution with one and two kink waves, and the M-shaped interaction with rogue and kink waves. Furthermore, we also segregated the rich spectrum of soliton solutions, such as W-shape, U-shape, periodic, dark, bright, combo, rational, exponential, mixed trigonometric, and hyperbolic soliton wave solutions inherent in the KP equation. Also, the model's modulation instability (MI) is evaluated using linear stability theory. In addition, we also analyze the bifurcation as well as the sensitivity analysis of the selected model, which is particularly important for dynamical models. We examine the synergistic application of these methodologies to the (2+1)-dimensional KP equation, intending to unravel its complex dynamics and discover new phenomena. Furthermore, we investigate the ramifications of these findings for a variety of physical systems, including shallow water waves and plasma physics. These findings will help gain a better knowledge of nonlinear wave phenomena and fresh insights into the dynamics of complex systems by combining the Hirota bilinear technique and the MSSE method.