Noncommutative Burgers Equation

Abstract

We present a noncommutative version of the Burgers equation which possesses the Lax representation and discuss the integrability in detail. We find a noncommutative version of the Cole-Hopf transformation and succeed in the linearization of it. The linearized equation is the (noncommutative) diffusion equation and exactly solved. We also discuss the properties of some exact solutions. The result shows that the noncommutative Burgers equation is completely integrable even though it contains infinite number of time derivatives. Furthermore, we derive the noncommutative Burgers equation from the noncommutative (anti-)self-dual Yang-Mills equation by reduction, which is an evidence for the noncommutative Ward conjecture. Finally, we present a noncommutative version of the Burgers hierarchy by both the Lax-pair generating technique and the Sato's approach.

arXiv comment24 pages, LaTeX, 1 figure; v2: discussions on Ward conjecture, Sato theory and the integrability added, references added, version to appear in J. Phys. A

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• Some properties of the integrable noncommutative sine-Gordon system
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• The U(N) chiral model and exact multi-solitons
• Burgers and KP hierarchies: A functional representation approach
• Matrix solutions of a noncommutative KP equation and a noncommutative mKP equation
• On non commutative sinh-Gordon Equation
• Moyal Noncommutative Integrability and the Burgers-KdV Mapping
• Noncommutative Instantons and Solitons
• Soliton Scattering in Noncommutative Spaces
• Quasideterminant solutions of NC Painlevé II equation with the Toda solution at $n=1$ as a seed solution in its Darboux transformation
• On Non-Commutative Integrable Burgers Equations
• Noncommutative Solitons in a Supersymmetric Chiral Model in 2+1 Dimensions
• Conserved Quantities in Noncommutative Principal Chiral Model with Wess-Zumino Term
• Scattering of Noncommutative Waves and Solitons in a Supersymmetric Chiral Model in 2+1 Dimensions
• About the self-dual Chern-Simons system and Toda field theories on the noncommutative plane
• Soliton equations: admitted solutions and invariances via Bäcklund transformations
• Chern-Simons Solitons, Chiral Model, and (affine) Toda Model on Noncommutative Space