Novel Soliton Propagation Governed by a Multi-Component Generalized NLS Equation with Fractional Order and Variable Coefficients

This study reveals novel soliton propagation governed by a variable-coefficient fractional-order multi-component generalized nonlinear Schrödinger (vfmgNLS) equation with the Riesz derivative, using an analytical approach. Specifically, the vfmgNLS equation with the Riesz derivative, abbreviated as the vRfmgNLS equation, is first mathematically derived, clarifying the anomalous dispersion relation and the corresponding operator function. Based on the transformation into a related Riemann–Hilbert (RH) problem with solvability, the N-soliton solutions of the vRfmgNLS equation are then accurately obtained. In particular, for the case where the multi-component is specifically limited to three, further analytical expressions combining fractional order and variable coefficients of the one-soliton solutions and two-soliton solutions are presented, and the novel propagation characteristics of these soliton solutions, including periodicity, symmetry, and reflection, are finally analyzed through three-dimensional images.