Solving the Generalized Nonlinear Schrِdinger Equation via Quartic Spline Approximation

This paper is concerned with a new conservative finite difference method for solving the generalized nonlinear Schrödinger (GNLS) equation iut+uxx+f(⊢u⊢2)u=0. The numerical scheme is constructed through the semidiscretization and an application of the quartic spline approximation. Central difference and extrapolation formulae are used for approximating the Neumann boundary conditions introduced. Both continuous and discrete energy conservation and the stability property are investigated. The numerical method provides an efficient and reliable way for computing long-time solitary solutions given by the GNLS equation. Numerical examples are given to demonstrate our conclusions.