A linear implicit finite difference discretization of the Schrodinger-Hirota Equation

A linear implicit finite difference method is proposed for the approximation of the solution to a periodic, initial value problem for a Schrodinger-Hirota equation. Optimal, second order convergence in the discrete $H^1-$norm is proved, assuming that $\tau$, $h$ and $\tau^4/h$ are sufficiently small, where $\tau$ is the time-step and $h$ is the space mesh-size. The efficiency of the proposed method is verified by results from numerical experiments.