Novel Combined Technique for Solving Chen–Lee–Liu Equation Modeling Quantum Particle Dynamics

This paper presents a novel numerical approach for solving the nonlinear Chen–Lee–Liu (CLL) equation, an important model in nonlinear optics and wave dynamics. The proposed method combines Bernstein polynomials with the Adomian Decomposition Method (ADM) to improve accuracy and computational efficiency. The spectral-like properties of Bernstein polynomials provide an effective approximation framework, while ADM efficiently handles nonlinearities without requiring linearization or perturbation techniques. The governing equation is transformed into a system of ordinary differential equations (ODEs) by approximating the solution components with Bernstein polynomials. ADM is then applied to iteratively solve for the unknown coefficients. Numerical simulations demonstrate the superiority of the method over traditional techniques, showing improved accuracy and reduced computational time. The study includes comparative results with the Haar wavelet method. Real-life applications in nonlinear optical wave propagation and soliton dynamics further highlight the significance of the proposed technique. The findings suggest that the spectral Bernstein–Adomian approach can be effectively applied to other nonlinear partial differential equations in various scientific and engineering domains.