Exact Traveling Wave Solutions of Extended Kadomtsev–Petviashvili Equation via Modified Rational Sine–Cosine and New Extended Direct Algebraic Methods

This paper investigates the (2 + 1)–dimensional extended Kadomtsev–Petviashvili equation (eKPE), importantly describing multidimensional nonlinear wave phenomena. Its exact traveling wave solutions are constructed using the modified rational sine–cosine method and the new extended direct algebraic method (NEDAM). The obtained solutions are expressed in diverse functional forms, including trigonometric, hyperbolic, rational, exponential, and complex-valued structures. Graphical representations of selected solutions are plotted to illustrate their physical characteristics, showing various nonlinear wave behaviors such as a singular periodic wave, a periodic wave solution, a kink soliton wave, and a singular kink-type solution. In addition, a linear stability analysis is performed to examine the modulation behavior of the model under different parameter choices. The results demonstrate that the proposed analytical approaches provide an efficient and systematic framework for generating a wide spectrum of exact solutions to the eKPE, thus enriching the understanding of its nonlinear wave dynamics.