Crank–Nicolson Method for the Chiral nonlinear Schrödinger Equation

In this paper, we develop a finite difference scheme based on the Crank–Nicolson method for solving the chiral nonlinear Schrödinger (CNLS) equation, which describes the dynamics of nonlinear wave propagation with chirality effects. The CNLS equation supports two types of progressive wave solutions: bright solitons and dark solitons. The proposed Crank–Nicolson scheme is implicit, unconditionally stable, and achieves second-order accuracy in both space and time. To evaluate the accuracy of the method, numerical results are compared with exact analytical soliton solutions. Numerical simulations are presented for the propagation of single bright and dark solitons. The results demonstrate that the Crank–Nicolson method accurately preserves soliton structures, making it an effective tool for studying the dynamics governed by the chiral nonlinear Schrödinger equation. The study demonstrates the effectiveness of the Crank–Nicolson method in capturing the dynamics of chiral nonlinear wave propagation and lays the foundation for further exploration of chiral effects in quantum and optical systems.