Towards Noncommutative Integrable Equations
Abstract
We study the extension of integrable equations which possess the Lax representations to noncommutative spaces. We construct various noncommutative Lax equations by the Lax-pair generating technique and the Sato theory. The Sato theory has revealed essential aspects of the integrability of commutative soliton equations and the noncommutative extension is worth studying. We succeed in deriving various noncommutative hierarchy equations in the framework of the Sato theory, which is brand-new. The existence of the hierarchy would suggest a hidden infinite-dimensional symmetry in the noncommutative Lax equations. We finally show that a noncommutative version of Burgers equation is completely integrable because it is linearizable via noncommutative Cole-Hopf transformation. These results are expected to lead to the completion of the noncommutative Sato theory.
Keywords
Cite
@article{arxiv.hep-th/0309265,
title = {Towards Noncommutative Integrable Equations},
author = {Masashi Hamanaka and Kouichi Toda},
journal= {arXiv preprint arXiv:hep-th/0309265},
year = {2008}
}
Comments
14 pages, LaTeX, talk given by K.T. at the fifth international conference on Symmetry in Nonlinear Mathematical Physics, Kiev, Ukraine, 23-29 June 2003; v2: accepted for publication in the proceedings, typos corrected, references added