4 research outputs found
Noncommutative Integrable Field Theories in 2d
We study the noncommutative generalization of (euclidean) integrable models
in two-dimensions, specifically the sine- and sinh-Gordon and the U(N)
principal chiral models. By looking at tree-level amplitudes for the
sinh-Gordon model we show that its na\"\i ve noncommutative generalization is
{\em not} integrable. On the other hand, the addition of extra constraints,
obtained through the generalization of the zero-curvature method, renders the
model integrable. We construct explicit non-local non-trivial conserved charges
for the U(N) principal chiral model using the Brezin-Itzykson-Zinn-Justin-Zuber
method.Comment: 18 pages, 1 figure; v2: references adde
Noncommutative Burgers Equation
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.Comment: 24 pages, LaTeX, 1 figure; v2: discussions on Ward conjecture, Sato
theory and the integrability added, references added, version to appear in J.
Phys.