On the Analytical and Numerical Solutions of the One-Dimensional Nonlinear Schrodinger Equation

In this paper, four compelling numerical approaches, namely, the split-step Fourier transform (SSFT), Fourier pseudospectral method (FPSM), Crank-Nicolson method (CNM), and Hopscotch method (HSM), are exhaustively presented for solving the 1D nonlinear Schrodinger equation (NLSE). The significance of this equation is referred to its notable contribution in modeling wave propagation in a plethora of crucial real-life applications such as the fiber optics field. Although exact solutions can be obtained to solve this equation, these solutions are extremely insufficient because of their limitations to only a unique structure under some limited initial conditions. Therefore, seeking high-performance numerical techniques to manipulate this well-known equation is our fundamental purpose in this study. In this regard, extensive comparisons of the proposed numerical approaches, against the exact solution, are conducted to investigate the benefits of each of them along with their drawbacks, targeting a broad range of temporal and spatial values. Based on the obtained numerical simulations via MATLAB, we extrapolated that the SSFT invariably exhibits the topmost robust potentiality for solving this equation. However, the other suggested schemes are substantiated to be consistently accurate, but they might generate higher errors or even consume more processing time under certain conditions.