4,107 research outputs found
Differential Calculi on Quantum Spaces determined by Automorphisms
If the bimodule of 1-forms of a differential calculus over an associative
algebra is the direct sum of 1-dimensional bimodules, a relation with
automorphisms of the algebra shows up. This happens for some familiar quantum
space calculi.Comment: 7 pages, Proceedings of XIIIth International Colloquium Integrable
Systems and Quantum Group
KdV soliton interactions: a tropical view
Via a "tropical limit" (Maslov dequantization), Korteweg-deVries (KdV)
solitons correspond to piecewise linear graphs in two-dimensional space-time.
We explore this limit.Comment: 10 pages, 4 figures, conference "Physics and Mathematics of Nonlinear
Phenomena 2013
Multicomponent Burgers and KP Hierarchies, and Solutions from a Matrix Linear System
Via a Cole-Hopf transformation, the multicomponent linear heat hierarchy
leads to a multicomponent Burgers hierarchy. We show in particular that any
solution of the latter also solves a corresponding multicomponent (potential)
KP hierarchy. A generalization of the Cole-Hopf transformation leads to a more
general relation between the multicomponent linear heat hierarchy and the
multicomponent KP hierarchy. From this results a construction of exact
solutions of the latter via a matrix linear system.Comment: 18 pages, 4 figure
Extension of Moyal-deformed hierarchies of soliton equations
Moyal-deformed hierarchies of soliton equations can be extended to larger
hierarchies by including additional evolution equations with respect to the
deformation parameters. A general framework is presented in which the extension
is universally determined and which applies to several deformed hierarchies,
including the noncommutative KP, modified KP, and Toda lattice hierarchy. We
prove a Birkhoff factorization relation for the extended ncKP and ncmKP
hierarchies. Also reductions of the latter hierarchies are briefly discussed.
Furthermore, some results concerning the extended ncKP hierarchy are recalled
from previous work.Comment: 15 pages, proceedings XI-th International Conference Symmetry Methods
in Physics (Prague, June 2004
Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations
We present a general solution-generating result within the bidifferential
calculus approach to integrable partial differential and difference equations,
based on a binary Darboux-type transformation. This is then applied to the
non-autonomous chiral model, a certain reduction of which is known to appear in
the case of the D-dimensional vacuum Einstein equations with D-2 commuting
Killing vector fields. A large class of exact solutions is obtained, and the
aforementioned reduction is implemented. This results in an alternative to the
well-known Belinski-Zakharov formalism. We recover relevant examples of
space-times in dimensions four (Kerr-NUT, Tomimatsu-Sato) and five (single and
double Myers-Perry black holes, black saturn, bicycling black rings)
Weakly nonassociative algebras, Riccati and KP hierarchies
It has recently been observed that certain nonassociative algebras (called
"weakly nonassociative", WNA) determine, via a universal hierarchy of ordinary
differential equations, solutions of the KP hierarchy with dependent variable
in an associative subalgebra (the middle nucleus). We recall central results
and consider a class of WNA algebras for which the hierarchy of ODEs reduces to
a matrix Riccati hierarchy, which can be easily solved. The resulting solutions
of a matrix KP hierarchy then determine (under a rank 1 condition) solutions of
the scalar KP hierarchy. We extend these results to the discrete KP hierarchy.
Moreover, we build a bridge from the WNA framework to the Gelfand-Dickey
formulation of the KP hierarchy.Comment: 16 pages, second version: LaTeX problem with L's in section 5
resolved, third version: example 2 in section 3 added, some minor
corrections, forth version: a few small changes and corrections. Proceedings
of the workshop Algebra, Geometry, and Mathematical Physics, Lund, October,
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