Space-Fractional Variable-Coefficient Toda Lattice Equation: Model Derivation and Inverse Scattering Transform

This paper systematically studies a space-fractional variable-coefficient Toda (sfvToda) lattice equation. The core innovation of this paper lies in incorporating both fractional power of the recursion operator and time-varying coefficients into the Toda lattice, as well as supplementing relevant theories. Firstly, based on the adjoint operator of a 2 × 2 matrix operator defined, a fractional Toda lattice hierarchy is presented. Three time-varying coefficients are introduced to combine the first three members of this hierarchy into the sfvToda lattice equation. Secondly, the orthogonality and completeness of squared eigenfunions related to the Toda lattice spectral problem (SP) are presented, and the explicit representations of the space-fractional Toda lattice hierarchy and the sfvToda lattice equation are derived. Thirdly, within the framework of inverse scattering transform (IST) theory, the linearization of the sfvToda lattice equation is achieved through scattering data, and explicit fractional variable-velocity N-soliton solutions are obtained. Finally, numerical simulations of single-soliton solutions, double-soliton solutions, and triple-soliton solutions with variable velocity are carried out by selecting appropriate coefficient functions. The results indicate that both the fractional extension and time-varying coefficients have significant regulatory effects on the spatiotemporal structure and propagation velocity of the obtained fractional soliton solutions.