Noncommutative Solitons and Quasideterminants

Abstract

We discuss extension of soliton theory and integrable systems to noncommutative spaces, focusing on integrable aspects of noncommutative anti-self-dual Yang-Mills equations. We give wide class of exact solutions by solving a Riemann-Hilbert problem for the Atiyah-Ward ansatz and present Backlund transformations for the G=U(2) noncommutative anti-self-dual Yang-Mills equations. We find that one kind of noncommutative determinants, quasideterminants, play crucial roles in the construction of noncommutative solutions. We also discuss reduction of a noncommutative anti-self-dual Yang-Mills equation to noncommutative integrable equations. This is partially based on collaboration with C. Gilson and J. Nimmo (Glasgow).

22 pages, LaTeX; v3: published version, invited article for Physica Scripta

References in corpus (12)

• On a direct approach to quasideterminant solutions of a noncommutative modified KP equation
• Quasideterminant solutions of a non-Abelian Toda lattice and kink solutions of a matrix sine-Gordon equation
• Quasideterminant solutions of a non-Abelian Hirota-Miwa equation
• Noncommutative Toda Chains, Hankel Quasideterminants And Painlev'e II Equation
• Quantisation of twistor theory by cocycle twist
• With a Cole-Hopf transformation to solutions of the noncommutative KP hierarchy in terms of Wronski matrices
• Noncommutative Solitons
• On solutions to the non-Abelian Hirota-Miwa equation and its continuum limits
• Twistor Theory and Differential Equations
• On non commutative sinh-Gordon Equation
• Bäcklund Transformations for Noncommutative Anti-Self-Dual Yang-Mills Equations
• Bäcklund Transformations and the Atiyah-Ward ansatz for Noncommutative Anti-Self-Dual Yang-Mills Equations

Cited by in corpus (6)

• Binary Darboux Transformation for the Sasa-Satsuma Equation
• Multivariate orthogonal polynomial and integrable systems
• Soliton Solutions of Noncommutative Anti-Self-Dual Yang-Mills Equations
• Soliton Scattering in Noncommutative Spaces
• Multi-Soliton Dynamics of Anti-Self-Dual Gauge Fields
• Soliton equations: admitted solutions and invariances via Bäcklund transformations