Solving coupled nonlinear Schrödinger equation using finite difference method and hybrid cubic B-spline collocation method

Abstract

Coupled Nonlinear Schrödinger (CNLS) equation is a second order nonlinear partial differential equation commonly related to nonlinear optical fiber. In this paper, CNLS equation is solved using Finite Difference Method (FDM) and Hybrid Cubic B-Spline collocation method (HCBM) with appropriate initial and boundary conditions. Theta-weighted scheme is applied to the equations and the nonlinear terms are linearized using Taylor series expansion. The temporal space is discretized by forward difference and for the spatial dimensions, central difference is applied for FDM while B-Spline functions are applied for HCBM. The HCBM is shown to be unconditionally stable using von Neumann stability analysis. To test the accuracy, a numerical example is discussed and the error norms are computed. The results obtained show that FDM and HCBM are reliable and easy to implement.