Numerical Solutions and Conservation Analysis of the Fractional Davey–Stewartson System Using a Crank–Nicolson Scheme

ABSTRACT The fractional generalized Davey–Stewartson System (FGDSS) is an extension of the classical Davey–Stewartson model, introducing Riesz fractional derivatives to capture nonlocal interactions and anomalous dispersion effects. Incorporating fractional operators enhances the physical realism of multidimensional wave descriptions, but it also poses substantial difficulties for numerical discretization and stability control. In this study, we develop a fully discrete finite‐difference scheme that employs the Crank–Nicolson (CN) approach for temporal integration together with Ortigueira's centered‐difference method for approximating Riesz fractional derivatives in space. The proposed algorithm attains second‐order accuracy in both time and space while maintaining a discrete counterpart of the Hamiltonian structure. To resolve the nonlinear coupling, we adopt an iterative Picard strategy at each time increment. A key contribution of this work is the demonstration that, under homogeneous Dirichlet boundary conditions, the method preserves discrete analogs of key invariants such as mass, energy, and momentum. Through extensive numerical tests—including scenarios featuring soliton dynamics—we validate the accuracy, robustness, and long‐time stability of the scheme. These results highlight the potential of the CN‐Ortigueira formulation as a practical tool for analyzing complex fractional dispersive models and pave the way for future developments involving adaptive meshes and higher‐order discretizations.