  A vectorial FDTD method is used to present a numerical study of very narrow
spatial solitons interacting with the surface of what has become known as a
left-handed medium. After a comprehensive discussion of the background and the
family of surface modes to be expected on a left-handed material, bounded by
dispersion-free right-handed material, it is demonstrated that robust outcomes
of the FDTD approach yield dramatic confirmation of these waves. The FDTD
results show how the linear and nonlinear surface modes are created and can be
tracked in time as they develop. It is shown how they can move backwards, or
forwards, depending either upon a critical value of the local nonlinear
conditions at the interface, or the ambient linear conditions. Several examples
are given to demonstrate the power and versatility of the method, and the
sensitivity to the launching conditions.
