Integrable Riesz Fractional-Order Generalized NLS Equation with Variable Coefficients: Inverse Scattering Transform and Analytical Solutions

Significant new progress has been made in nonlinear integrable systems with Riesz fractional-order derivative, and it is impressive that such nonlocal fractional-order integrable systems exhibit inverse scattering integrability. The focus of this article is on extending this progress to nonlocal fractional-order Schrödinger-type equations with variable coefficients. Specifically, based on the analysis of anomalous dispersion relation (ADR), a novel variable-coefficient Riesz fractional-order generalized NLS (vcRfgNLS) equation is derived. By utilizing the relevant matrix spectral problems (MSPs), the vcRfgNLS equation is solved through the inverse scattering transform (IST), and analytical solutions including n-soliton solution as a special case are obtained. In addition, an explicit form of the vcRfgNLS equation depending on the completeness of squared eigenfunctions (SEFs) is presented. In particular, the 1-soliton solution and 2-soliton solution are taken as examples to simulate their spatial structures and analyze their structural properties by selecting different variable coefficients and fractional orders. It turns out that both the variable coefficients and fractional order can influence the velocity of soliton propagation, but there is no energy dissipation throughout the entire motion process. Such soliton solutions may not only have important value for studying the super-dispersion transport of nonlinear waves in non-uniform media, but also for realizing a new generation of ultra-high-speed optical communication engineering.