Computational analysis of the conformable nonlinear Schrödinger equation with Kudryashov’s refractive index model and generalized non-local nonlinearity

This paper investigates the nonlinear conformable Schrödinger equation with nonlocal nonlinearities and Kudryashov’s arbitrary refractive index. The study employs the Uniform Method (UM) and the Extended Hyperbolic Function Method (EHFM) to derive various optical soliton solutions for this present conformable equation. To demonstrate the importance of the novel solutions, the paper presents two-dimensional, three-dimensional, and contour plots, illustrating kink-type, wave, bright, bell-shaped, and mixed dark–bright soliton solutions. Further, the impact of the fractional order parameter, temporal parameter, and nonlinearity coefficient on these optical solutions is analyzed, providing valuable insights into the conformable nonlinear Schrödinger model. The behavior of these optical solutions is analyzed through illustrative graphs that account for variations in the conformable order derivative, temporal parameter, and nonlinearity coefficient. The methodologies applied in this research show potential for broader application to various nonlinear Schrödinger equations in fields such as applied mathematics and nonlinear optics. The research suggests that these methods could be applied in the future to investigate other differential equations with fractional and integer orders across different fields of applied sciences. This study expands our understanding of nonlinear optics and has potential practical implications in various fields like optical signal processing, laser technology, and telecommunications.