Exact Noncommutative KP and KdV Multi-solitons

TL;DR

This paper derives exact multi-soliton solutions for noncommutative KP and KdV equations, revealing how noncommutativity affects soliton interactions and the tau function construction.

Contribution

It introduces a method to construct N-soliton solutions for noncommutative KP and KdV equations, including cases with arbitrary space-space noncommutativity.

Findings

Multi-soliton solutions exist for noncommutative KP and KdV equations.

Noncommutativity obstructs the construction of a tau function.

Asymptotic soliton scattering appears unaffected by noncommutativity.

Abstract

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.