Exact Soliton Solutions of Space-Time Fractional Schrödinger Equations with Kerr and Cubic-Quintic Nonlinearities via the Sine-Gordon Expansion Method

In this study, we investigate analytical soliton solutions of space-time fractional nonlinear Schrödinger equations (FNLSEs) incorporating Kerr law and cubic-quintic nonlinearities. These equations play a fundamental role in modeling a wide range of physical phenomena, including nonlinear plasma physics, optics, and Bose-Einstein condensation. By employing the sine-Gordon expansion approach, a wide variety of soliton solutions are derived, such as periodic, modulated periodic, kink, anti-kink, V-shaped, M-shaped soliton, and wave packet-like structures, and these results extend the solution space in the complex domain. The solutions are formulated using exponential, hyperbolic, and trigonometric functions. The conformable fractional derivative is utilized to establish a consistent and generalized mathematical framework. The influence of key physical parameters such as wave frequency, fractional order, and wave number on soliton behavior is systematically analyzed. The findings confirm the effectiveness of the SGE method in producing exact solutions and underline its potential for application in diverse scientific domains, including quantum engineering, nonlinear optics, and fluid dynamics.