A Linear Implicit Finite Difference Discretization of the Schrödinger–Hirota Equation

A new linear implicit finite difference method is proposed for the approximation of the solution to a periodic, initial value problem for a Schrödinger–Hirota equation. Optimal, second order convergence in the discrete H 1 -norm is proved, assuming that τ , h and τ 4 h are sufficiently small, where τ is the time-step and h is the space mesh-size. The convergence analysis is based on the investigation of a modified version of the proposed finite difference method, which is innovative and handles the stability difficulties due to the presence of a nonlinear derivative term in the equation. The efficiency of the proposed finite difference method is verified by results from numerical experiments.