Retrieval of optical soliton patterns for time-fractional nonlinear Schrödinger equation integrating Kudryashov’s law of refractive index with dual form of nonlocal nonlinearity

The Schrödinger model, which describes the mobility of a single wave in an optical fiber, is crucial to communication systems. By using the quadrupled-power law and the dual form of nonlocal nonlinearity, this study concentrates on establishing optical solitary solutions for Kudryashov’s law of nonlinear refractive index. The unified Riccati equation method and the new extended auxiliary equation approach are used to derive numerous ranges of Jacobi elliptic, trigonometric, hyperbolic, and dual-wave solitary solution patterns. These findings will eventually lead to the discovery of bright, singular, kink, dark–bright, and singular-periodic soliton solutions, whose 3D, 2D, and density depictions are provided with the help of mathematical software, Mathematica. Moreover, we provide 2D graphical representations for different values of the fractional and time parameters to show how these novel optical solutions behave. Different features of these solitons arise in many physical contexts such as nonlinear optics, fluid dynamics, and laser physics. The derived results show that the proposed framework presents a rich and varied spectrum of wave phenomena and that the methods for solving nonlinear partial differential equations arising in mathematical physics are effective, useful, and dependable for further exploration in the vast field of optical and mathematical physics.