CONSTRUCTING SOLITON WAVE STRUCTURES FOR THE NONLINEAR FRACTIONAL COMPLEX GINZBURG–LANDAU EQUATION AND ITS STABILITY ANALYSIS USING AN ANALYTICAL APPROACH

This work makes a significant contribution by providing a comprehensive study of exact solutions of the fractional complex Ginzburg–Landau equation based on the conformable fractional derivative. Fractional derivatives bring intrinsic non-locality to the theory that accommodates memory effects and complex temporal dynamics absent from the integer-order classical formulation. As a simple nonlinear equation, the complex Ginzburg–Landau equation plays a central role in describing many physical phenomena, including superconductivity, superfluidity, nonlinear optics, and pattern formation. The novelty in this work is to extend the modified direct algebraic method to the fraction form of the equation to generate a wide variety of exact solutions from dark, bright, and singular solitons to Jacobi elliptic, exponential, singular periodic, and rational ones. These diverse waveforms provide new analytical insight into the broad solution space of the fractional system. For a complete linear stability analysis, the stability of solutions so obtained is studied with respect to small amplitude perturbations to ensure their physical validity. Graphical representations are also given to present the dynamic development of these solutions for various orders of fractional derivative, revealing the wide influence of fractional effects on wave profiles. These findings not only make the list of solutions to known fractional Ginzburg–Landau systems richer but also pave the way for their application in nonlinear complex physics and engineering phenomena modeling in modern physics and engineering.