334 research outputs found
Extension of Noncommutative Soliton Hierarchies
A linear system, which generates a Moyal-deformed two-dimensional soliton
equation as integrability condition, can be extended to a three-dimensional
linear system, treating the deformation parameter as an additional coordinate.
The supplementary integrability conditions result in a first order differential
equation with respect to the deformation parameter, the flow of which commutes
with the flow of the deformed soliton equation. In this way, a deformed soliton
hierarchy can be extended to a bigger hierarchy by including the corresponding
deformation equations. We prove the extended hierarchy properties for the
deformed AKNS hierarchy, and specialize to the cases of deformed NLS, KdV and
mKdV hierarchies. Corresponding results are also obtained for the deformed KP
hierarchy. A deformation equation determines a kind of Seiberg-Witten map from
classical solutions to solutions of the respective `noncommutative' deformed
equation.Comment: 19 pages, some minor changes in section 4, typos in
(2.4),(3.38),(4.15) corrected, to appear in Journal of Physics
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,
200
Algebraic identities associated with KP and AKNS hierarchies
Explicit KP and AKNS hierarchy equations can be constructed from a certain
set of algebraic identities involving a quasi-shuffle product.Comment: 6 pages, proceedings of Integrable Systems 2005, Pragu
Simplex and Polygon Equations
It is shown that higher Bruhat orders admit a decomposition into a higher
Tamari order, the corresponding dual Tamari order, and a "mixed order." We
describe simplex equations (including the Yang-Baxter equation) as realizations
of higher Bruhat orders. Correspondingly, a family of "polygon equations"
realizes higher Tamari orders. They generalize the well-known pentagon
equation. The structure of simplex and polygon equations is visualized in terms
of deformations of maximal chains in posets forming 1-skeletons of polyhedra.
The decomposition of higher Bruhat orders induces a reduction of the
-simplex equation to the -gon equation, its dual, and a compatibility
equation
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
From the Kadomtsev-Petviashvili equation halfway to Ward's chiral model
The "pseudodual" of Ward's modified chiral model is a dispersionless limit of
the matrix Kadomtsev-Petviashvili (KP) equation. This relation allows to carry
solution techniques from KP over to the former model. In particular, lump
solutions of the su(m) model with rather complex interaction patterns are
reached in this way. We present a new example.Comment: 6 pages, 2 figures, Workshop "Algebra, Geometry, and Mathematical
Physics", Goeteborg, October 2007, 2nd version: corrections on page
Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions
We express AKNS hierarchies, admitting reductions to matrix NLS and matrix
mKdV hierarchies, in terms of a bidifferential graded algebra. Application of a
universal result in this framework quickly generates an infinite family of
exact solutions, including e.g. the matrix solitons in the focusing NLS case.
Exploiting a general Miura transformation, we recover the generalized
Heisenberg magnet hierarchy and establish a corresponding solution formula for
it. Simply by exchanging the roles of the two derivations of the bidifferential
graded algebra, we recover "negative flows", leading to an extension of the
respective hierarchy. In this way we also meet a matrix and vector version of
the short pulse equation and also the sine-Gordon equation. For these equations
corresponding solution formulas are also derived. In all these cases the
solutions are parametrized in terms of matrix data that have to satisfy a
certain Sylvester equation
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