  We investigate numerically the coupled diffusion-advective type field
equations originating from the canonical phase space approach to the noisy
Burgers equation or the equivalent Kardar-Parisi-Zhang equation in one spatial
dimension. The equations support stable right hand and left hand solitons and
in the low viscosity limit a long-lived soliton pair excitation. We find that
two identical pair excitations scatter transparently subject to a size
dependent phase shift and that identical solitons scatter on a static soliton
transparently without a phase shift. The soliton pair excitation and the
scattering configurations are interpreted in terms of growing step and
nucleation events in the interface growth profile. In the asymmetrical case the
soliton scattering modes are unstable presumably toward multi soliton
production and extended diffusive modes, signalling the general
non-integrability of the coupled field equations. Finally, we have shown that
growing steps perform anomalous random walk with dynamic exponent z=3/2 and
that the nucleation of a tip is stochastically suppressed with respect to
plateau formation.
