  We derive the Kadomtsev-Petviashvili (KP) equation defined over a general
associative algebra and construct its N-soliton solution. For the example of
the Moyal algebra, we find multi-soliton solutions for arbitrary space-space
noncommutativity. The noncommutativity of coordinates is shown to obstruct the
general construction of a tau function for these solitons. We investigate the
two-soliton solution in detail and show that asymptotic observers of soliton
scattering are unable to detect a finite spatial noncommutativity. An explicit
example shows that a pair of solitons in a noncommutative background can be
interpreted as several pairs of image solitons. Finally, a dimensional
reduction gives the general N-soliton solution for the previously discussed
noncommutative KdV equation.
