  An initial value problem of the one-dimensional nonlinear Schr\"odinger (NLS)
equation with constant dispersive and nonlinear coefficients can be solved
using a compact finite difference scheme (Xie, Li, & Yi, 2009). A similar
scheme is implemented in the signalling problem of the one-dimensional NLS
equation with constant coefficient where it describes the propagation of
surface gravity wave packet over a flat bottom. Various examples are
illustrated. A similar compact finite difference scheme is modified and
implemented further in the signalling problem of the one-dimensional NLS
equation with variable dispersive and nonlinear coefficients which models the
surface wave packet propagation over the slowly varying bottom. Several
topographies are considered and illustrated. It is observed that the wave
envelope signal propagates opposed to topography height. The possibility of
shaping topography near shores area to minimise the effect of high amplitude
solitary waves are also examined.
