Bäcklund Transformations and the Atiyah-Ward ansatz for Noncommutative Anti-Self-Dual Yang-Mills Equations

Abstract

We present Backlund transformations for the noncommutative anti-self-dual Yang-Mills equation where the gauge group is G=GL(2) and use it to generate a series of exact solutions from a simple seed solution. The solutions generated by this approach are represented in terms of quasideterminants. We also explain the origins of all of the ingredients of the Backlund transformations within the framework of noncommutative twistor theory. In particular we show that the generated solutions belong to a noncommutative version of the Atiyah-Ward ansatz.

v2: 21 pages, published version

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• Quasideterminant solutions of a non-Abelian Toda lattice and kink solutions of a matrix sine-Gordon equation
• Noncommutative Solitons and D-branes
• Quantisation of twistor theory by cocycle twist
• Noncommutative Solitons
• Noncommutative Supersymmetric/Integrable Models and String Theory
• Bäcklund Transformations for Noncommutative Anti-Self-Dual Yang-Mills Equations
• Topics in noncommutative integrable theories and holographic brane-world cosmology