Analytical construction of needle-type solitons in a M-fractional paraxial wave framework with dynamical analysis.

In this paper, to investigate exact optical soliton solutions and their dynamical properties for the paraxial wave model involving the M-fractional derivative. To achieve this, employ the improved modified Sardar sub-equation method, which provides a systematic and efficient analytical framework for constructing closed-form solutions of nonlinear fractional wave equations. The resulting fractional complex paraxial wave dynamical (FPWD) model is of considerable importance due to its wide applicability in nonlinear optics, optical fiber communications, quantum electronics, and plasma physics. The mathematical analysis reveals the existence of new solitary wave solutions that demonstrate complex nonlinear patterns that govern the system. The research discovers a needle-type soliton structure that operates within an M-fractional paraxial wave framework, thus differentiating itself from existing research, which uses classical and other fractional operators. The research displays three-dimensional surface plots and contour maps to enhance understanding of the obtained wave structures. The study assesses how essential parameters affect solution stabilityAQ through a sensitivity analysis, which shows solitons maintain their strength and operational behavior across various physical conditions.