6 research outputs found
Notes on Exact Multi-Soliton Solutions of Noncommutative Integrable Hierarchies
We study exact multi-soliton solutions of integrable hierarchies on
noncommutative space-times which are represented in terms of quasi-determinants
of Wronski matrices by Etingof, Gelfand and Retakh. We analyze the asymptotic
behavior of the multi-soliton solutions and found that the asymptotic
configurations in soliton scattering process can be all the same as commutative
ones, that is, the configuration of N-soliton solution has N isolated localized
energy densities and the each solitary wave-packet preserves its shape and
velocity in the scattering process. The phase shifts are also the same as
commutative ones. Furthermore noncommutative toroidal Gelfand-Dickey hierarchy
is introduced and the exact multi-soliton solutions are given.Comment: 18 pages, v3: references added, version to appear in JHE
On a direct approach to quasideterminant solutions of a noncommutative KP equation
A noncommutative version of the KP equation and two families of its solutions
expressed as quasideterminants are discussed. The origin of these solutions is
explained by means of Darboux and binary Darboux transformations. Additionally,
it is shown that these solutions may also be verified directly. This approach
is reminiscent of the wronskian technique used for the Hirota bilinear form of
the regular, commutative KP equation but, in the noncommutative case, no
bilinearising transformation is available.Comment: 11 page
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.