Exact Noncommutative KP and KdV Multi-solitons

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 τ 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.