Optical solitons with polynomial law of self-phase modulation by the fourth-order conservative finite difference scheme

This paper considers an initial-boundary value problem for optical soliton propagation based on the cubic-quintic-septic nonlinear Schrödinger equation. We first analyzed the underlying physical properties of the system, formally establishing the conservation laws for mass and energy. Subsequently, we proposed a fourth-order finite difference scheme derived via the discrete variational derivative method to effectively approximate the nonlinear potential. We rigorously proved that the resulting numerical scheme perfectly satisfies the discrete analogues of the mass and energy conservation laws. Furthermore, we provided comprehensive stability and convergence analysis, demonstrating unconditional stability and establishing an error bound of $O({\tau }^2+h^4)$. Numerical experiments were conducted to validate the theoretical analysis and illustrate the efficiency and high-order accuracy of the proposed approach.