Towards Noncommutative Integrable Systems

Abstract

We present a powerful method to generate various equations which possess the Lax representations on noncommutative (1+1) and (1+2)-dimensional spaces. The generated equations contain noncommutative integrable equations obtained by using the bicomplex method and by reductions of the noncommutative (anti-)self-dual Yang-Mills equation. This suggests that the noncommutative Lax equations would be integrable and be derived from reductions of the noncommutative (anti-)self-dual Yang-Mills equation, which implies the noncommutative version of Richard Ward conjecture. The integrability and the relation to string theories are also discussed.

13 pages, 1 figure, LaTeX; v2: typos corrected, comments and references added; v3: comments and references added, version to appear in Phys. lett. A

References in corpus (13)

• Noncommutative Field Theory
• Komaba Lectures on Noncommutative Solitons and D-Branes
• Noncommutative Multi-Solitons in 2+1 Dimensions
• Trieste lectures on solitons in noncommutative gauge theories
• Scattering of Noncommutative Solitons in 2+1 Dimensions
• Noncommutative Solitons in Open N=2 String Theory
• Exact Noncommutative KP and KdV Multi-solitons
• Noncommutative 't Hooft Instantons
• Open N=2 strings in a B-field background and noncommutative self-dual Yang-Mills
• Noncommutative Instantons via Dressing and Splitting Approaches
• Noncommutative Generalized NS and Super Matrix KdV Systems from a Noncommutative Version of (Anti-) Self-Dual Yang-Mills Equations
• Bicomplexes and Backlund transformations
• Burgers' equation in non-commutative space-time

Cited by in corpus (46)

• On a direct approach to quasideterminant solutions of a noncommutative modified KP equation
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• Quasideterminant solutions of a non-Abelian Hirota-Miwa equation
• Noncommutative Burgers Equation
• Noncommutative Solitons and D-branes
• Noncommutative Ward's Conjecture and Integrable Systems
• Noncommutative Extended Waves and Soliton-like Configurations in N=2 String Theory
• Integrable Discretisations for a Class of Nonlinear Schrodinger Equations on Grassmann Algebras
• Extension of Noncommutative Soliton Hierarchies
• Commuting Flows and Conservation Laws for Noncommutative Lax Hierarchies
• Darboux transformations for a twisted derivation and quasideterminant solutions to the super KdV equation
• Some properties of the integrable noncommutative sine-Gordon system
• On the structure of $(2+1)$--dimensional commutative and noncommutative integrable equations
• On Reductions of Noncommutative Anti-Self-Dual Yang-Mills Equations
• Notes on Exact Multi-Soliton Solutions of Noncommutative Integrable Hierarchies
• On solutions to the non-Abelian Hirota-Miwa equation and its continuum limits
• Towards Noncommutative Integrable Equations
• Noncommutative Inverse Scattering Method for the Kontsevich system
• On the $3D$ consistency of a Grassmann extended lattice Boussinesq system
• Time-Space Noncommutative Abelian Solitons
• Noncommutative Solitons and Quasideterminants
• Bäcklund Transformations and Non-Abelian Nonlinear Evolution Equations: a Novel Bäcklund Chart
• Soliton Solutions of Noncommutative Anti-Self-Dual Yang-Mills Equations
• Symmetries and reductions on the noncommutative Kadomtsev-Petviashvili and Gelfand-Dickey hierarchies
• Non-commutative solitons and strong-weak duality
• Noncommutative Self-Dual Supersymmetric Yang-Mills Theory
• Non-integrability of Self-dual Yang-Mills-Higgs System
• Moyal Noncommutative Integrability and the Burgers-KdV Mapping
• On non commutative sinh-Gordon Equation
• Quasideterminant solutions of NC Painlevé II equation with the Toda solution at $n=1$ as a seed solution in its Darboux transformation
• Soliton Scattering in Noncommutative Spaces
• Abelian Toda field theories on the noncommutative plane
• Multi-Soliton Dynamics of Anti-Self-Dual Gauge Fields
• Noncommutative Solitons in a Supersymmetric Chiral Model in 2+1 Dimensions
• Lax pair and Darboux transformation of noncommutative U(N) principal chiral model
• Explicit Flow Equations and Recursion Operator of the ncKP hierarchy
• Conserved Quantities in Noncommutative Principal Chiral Model with Wess-Zumino Term
• Solitons in Open N=2 String Theory
• 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
• Quasideterminant solutions to a noncommutative $q$-difference two-dimensional Toda lattice equation
• Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy
• Lax pair representation and Darboux transformation of NC Painlevé-II equation
• Darboux solutions of non-abelian quantum Painlevé II equation in terms of quasideterminants
• Noncommutative Root Space Witt, Ricci Flow, and Poisson Bracket Continual Lie Algebras
• Chern-Simons Solitons, Chiral Model, and (affine) Toda Model on Noncommutative Space