Stability analysis and exploration of multiform soliton solutions for extended fractional NLS model using modified extended direct algebraic method

Abstract

Abstract This paper investigates a generalized nonlinear Schrödinger-type equation involving fractional derivatives in both space and time, formulated through the recently introduced $$\beta$$ -fractional operator. The modified extended direct algebraic method (MEDM) is applied to derive new classes of exact analytical solutions, including bright, dark, periodic, and singular solitons. A detailed analysis of the fractional parameters reveals their quantitative influence on soliton width, velocity, and localization. The modulation instability (MI) gain spectra are evaluated to identify stability regions and illustrate how decreasing fractional orders suppress instability growth. Physically, the results demonstrate that the fractional orders act as tunable parameters governing wave dispersion, nonlocality, and energy localization in nonlinear media. The study establishes a flexible framework for controlling soliton dynamics in optical and plasma systems, underscoring the fundamental role of fractional calculus in modeling complex nonlinear wave phenomena.