Exact Noncommutative KP and KdV Multi-solitons

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.

18 pages LaTeX. Reference added

References in corpus (10)

• Noncommutative Perturbative Dynamics
• Noncommutative Solitons
• Descent Relations Among Bosonic D-branes
• Noncommutative Tachyons
• Komaba Lectures on Noncommutative Solitons and D-Branes
• Noncommutative Solitons in Open N=2 String Theory
• Noncommutative Solitons: Moduli Spaces, Quantization, Finite Theta Effects and Stability
• Exact solitons on noncommutative tori
• Noncommutative Korteweg-de-Vries Equation
• Noncommutative Generalized NS and Super Matrix KdV Systems from a Noncommutative Version of (Anti-) Self-Dual Yang-Mills Equations

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• Noncommutative Ward's Conjecture and Integrable Systems
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