Geometrical representation and Hirota direct approach for multiple soliton solutions of nonlinear M-coupled fractional equations

This paper introduces a novel analytical framework for deriving multiple soliton and singular soliton solutions to M-coupled fractional evolution equations. By integrating conformable fractional derivatives with an extended Hirota direct method, we systematically solve fractional versions of the KdV, mKdV, KP, and modified KP equations. The conformable derivative permits effective bilinearization, facilitating the construction of explicit solutions. We further provide a geometric interpretation through curvature analysis of soliton surfaces in fractional space. Theoretical results are validated against classical cases (α = 1), demonstrating consistency and enhancing the analytical toolkit for modeling wave propagation in nonlinear optics, plasma physics, and anomalous diffusion.