(untitled) (European Journal of Pure and Applied Mathematics, 2026)
This paper derives novel exact elliptic solutions for the (2+1)-dimensional Kadomtsev-Petviashvili equation and the Shallow Water Waves equation with beta-fractional derivative using the improved modified extended (IME) tanh method. The originality of the paper resides in the fact that the IME method was generalized for the fractional dispersive equations for the first time, resulting in a wide variety of new wave patterns, namely the bright and dark solitons, singular solitons, Jacobi elliptic waves, periodic, singular periodic, and hyperbolic solutions. The applicability of the obtained results is justified by the fact that these wave patterns are capable of characterizing the nonlocal wave processes occurring in shallow water and plasma-like media. Additionally, a bifurcation and stability analysis was also carried out, and the effects of the most significant parameters of the system on wave phenomena and transfer processes have been clarified. The obtained results are verified and illustrated using detailed 2D, 3D, and polar plots along with the phase portrait analysis.