Novel Breather, Lump and Interaction Solutions for (2+1)-dimensional Generalized Korteweg-De Vries Equation

The integrability of the (2+1)-dimensional generalized Korteweg-De Vries equation is examined in this work. In many areas of engineering and science, the governing equation is used, particularly in the investigation of nonlinear wave phenomena. This equation is appropriate for a wider variety of wave phenomena, such as solitons in optical fibers, ion-acoustic waves in plasma, and shallow water waves, since it takes into account higher-order nonlinear and dispersive effects. The study employs the Hirota bilinear method and focuses on certain Ansatz transformations and symbolic computation approaches to generate lump waves, rogue waves, breather waves, multi-wave solutions, and lump and kink wave combinations for the given problem. The technique demonstrates its adaptability in handling various wave events within nonlinear systems by methodically deriving a range of intricate wave patterns by using these transformations. A comprehensive analysis of the dynamics and distinctive features of the derived solutions is conducted through computational simulations, which utilize graphical representations and pay particular attention to specific parameter values. Additionally, the extended transformed rational function method is applied to the Hirota bilinear form of the (2+1)-dimensional generalized Korteweg-De Vries equation to extract complexiton solutions. The dynamics of these solutions are depicted through 3D, 2D, and contour graphics, which help elucidate the unique behaviors and characteristics of the solutions obtained.