Various traveling wave solutions for (2+1)-dimensional extended Kadomtsev–Petviashvili equation using a newly created methodology

The Kadomtsev–Petviashvili (KP) equation is a crucial model in various physical systems, including hydrodynamic wave disturbances, plasma physics, and nonlinear optics. Our study focuses on the analytical solutions of the (2+1)-dimensional extended KP equation, as these solutions can offer mathematical tools for understanding wave behavior and have practical applications. We found that incorporating two additional terms can restore the integrability of the equation, leading to the generation of dark and bright soliton and traveling wave solutions. In this work, we employ the Kumar-Malik method to find analytical solutions to the (2+1)-dimensional extended KP equation. The Kumar-Malik method is an effective approach for solving nonlinear partial differential equations (NLPDEs) based on a first-order differential equation. By applying this method, we have derived various solutions to the (2+1)-dimensional extended KP equation, including Jacobi elliptic, hyperbolic, trigonometric, and exponential function solutions. These solutions are then presented graphically to illustrate the wave behavior under different parameters. Our results contribute to a deeper understanding of the KP equation’s behavior under different physical conditions. Specifically, we have examined the effects of parameters on the widths, velocities, and other essential properties of the waves. This information is invaluable for studying hydrodynamic waves, plasma fluctuations, and optical solitons. In conclusion, this work underscores the importance of obtaining analytical solutions to the (2+1)-dimensional extended KP equation and presenting these solutions graphically. These solutions provide a valuable resource for understanding the complex behavior of physical systems and can inspire future research.