165 research outputs found
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
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
Discrete differential calculus, graphs, topologies and gauge theory
Differential calculus on discrete sets is developed in the spirit of
noncommutative geometry. Any differential algebra on a discrete set can be
regarded as a `reduction' of the `universal differential algebra' and this
allows a systematic exploration of differential algebras on a given set.
Associated with a differential algebra is a (di)graph where two vertices are
connected by at most two (antiparallel) arrows. The interpretation of such a
graph as a `Hasse diagram' determining a (locally finite) topology then
establishes contact with recent work by other authors in which discretizations
of topological spaces and corresponding field theories were considered which
retain their global topological structure. It is shown that field theories, and
in particular gauge theories, can be formulated on a discrete set in close
analogy with the continuum case. The framework presented generalizes ordinary
lattice theory which is recovered from an oriented (hypercubic) lattice graph.
It also includes, e.g., the two-point space used by Connes and Lott (and
others) in models of elementary particle physics. The formalism suggests that
the latter be regarded as an approximation of a manifold and thus opens a way
to relate models with an `internal' discrete space ({\`a} la Connes et al.) to
models of dimensionally reduced gauge fields. Furthermore, also a `symmetric
lattice' is studied which (in a certain continuum limit) turns out to be
related to a `noncommutative differential calculus' on manifolds.Comment: 36 pages, revised version, appendix adde
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)
Matrix KP: tropical limit and Yang-Baxter maps
We study soliton solutions of matrix Kadomtsev-Petviashvili (KP) equations in
a tropical limit, in which their support at fixed time is a planar graph and
polarizations are attached to its constituting lines. There is a subclass of
"pure line soliton solutions" for which we find that, in this limit, the
distribution of polarizations is fully determined by a Yang-Baxter map. For a
vector KP equation, this map is given by an R-matrix, whereas it is a
non-linear map in case of a more general matrix KP equation. We also consider
the corresponding Korteweg-deVries (KdV) reduction. Furthermore, exploiting the
fine structure of soliton interactions in the tropical limit, we obtain a new
solution of the tetrahedron (or Zamolodchikov) equation. Moreover, a solution
of the functional tetrahedron equation arises from the parameter-dependence of
the vector KP R-matrix.Comment: 23 pages, 9 figures, second version: some minor amendments,
reformulations in Section 4, additional references [10] and [18
A vectorial binary Darboux transformation of the first member of the negative part of the AKNS hierarchy
Using bidifferential calculus, we derive a vectorial binary Darboux
transformation for the first member of the "negative" part of the AKNS
hierarchy. A reduction leads to the first "negative flow" of the NLS hierarchy,
which in turn is a reduction of a rather simple nonlinear complex PDE in two
dimensions, with a leading mixed third derivative. This PDE may be regarded as
describing geometric dynamics of a complex scalar field in one dimension, since
it is invariant under coordinate transformations in one of the two independent
variables. We exploit the correspondingly reduced vectorial binary Darboux
transformation to generate multi-soliton solutions of the PDE, also with
additional rational dependence on the independent variables, and on a plane
wave background. This includes rogue waves.Comment: 19 pages, 5 figures, Second version: substantial changes. Third
version: Section 3 substantially expanded. Fourth version: small amendments
in Abstract, Introduction, first part of Section 3, and Conclusion. These
take a comment by Sakovich, arXiv:2205.09538v1 [nlin.SI], into accoun
Matrix Boussinesq solitons and their tropical limit
We study soliton solutions of matrix "good" Boussinesq equations, generated
via a binary Darboux transformation. Essential features of these solutions are
revealed via their "tropical limit", as exploited in previous work about the KP
equation. This limit associates a point particle interaction picture with a
soliton (wave) solution.Comment: 24 pages, 11 figures, second version: some minor amendment
Bicomplexes, Integrable Models, and Noncommutative Geometry
We discuss a relation between bicomplexes and integrable models, and consider
corresponding noncommutative (Moyal) deformations. As an example, a
noncommutative version of a Toda field theory is presented.Comment: 6 pages, 1 figure, LaTeX using amssymb.sty and diagrams.sty, to
appear in Proceedings of the 1999 Euroconference "Noncommutative geometry and
Hopf algebras in Field Theory and Particle Physics
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