Fractional and Integrable Perspectives on a Variable-Coefficient Semi-Discrete NLS Equation: Analytical Method and Engineering Applications

The method of deriving fractional-order differential equations using Riesz fractional-order calculus is a new breakthrough, and it is possible to use the inverse scattering transform (IST) to solve the analytical solutions of the derived equations. However, there are still relatively few examples of using discrete Riesz fractional (DRF) order calculations. This article focuses on using discrete Riesz fractional-order calculations to derive a variable-coefficient fractional integrable semi-discrete nonlinear Schrödinger (vcfISDNLS) equation. On the one hand, we derive the vcfISDNLS equation through dispersion relation (DR) and DRF order calculations. On the other hand, we obtain the explicit expressions of the one-soliton solution, two-soliton solution, and three-soliton solution of this equation without reflection potentials (RPs). By deriving and solving the equation and displaying the obtained soliton solutions, possible evidential support can be provided for control engineering, automotive engineering, image processing, and optical communication systems.