Dynamics of Lump, Breather, Two-Waves and Other Interaction Solutions of (2+1)-Dimensional KdV Equation

Abstract

In this investigation, we address a particular variant of the Korteweg–de Vries (KdV) equation, specifically focusing on the (2+1)-dimensional KdV equation. The equation can model various physical phenomena in different fields, including fluid dynamics, plasma physics, nonlinear optics, and other areas where coupled wave interactions are important. To commence, we establish the Auto-Bäcklund and Cole–Hopf transformations for the given model, resulting in the derivation of numerous soliton-like solutions characterized by hyperbolic, trigonometric, and exponential function waves. Furthermore, we effectively elucidate the behavior of lump, lump–kink, breather, two-wave, and three-wave solutions using the Hirota bilinear technique. Extensive numerical simulations employing 3-D profiles are conducted with meticulous consideration of pertinent parameter values, providing additional insights into the distinctive traits of the obtained solutions. Moreover, employing the extended transformed rational function method grounded in the bilinear form of the underlying equation, we uncover complexiton solutions. These solutions are depicted using 3-D and 2-D visualizations to portray their dynamics. Our findings reveal that the approach adopted to derive analytical solutions for nonlinear partial differential equations proves to be both efficient and potent. The combination of numerical simulations and visual representations enhances our understanding of these solutions, ultimately affirming the effectiveness and robustness of the employed methodology in tackling nonlinear partial differential equations.