522 research outputs found
Multilinear Operators: The Natural Extension Of Hirota's Bilinear Formalism
We introduce multilinear operators, that generalize Hirota's bilinear
operator, based on the principle of gauge invariance of the functions.
We show that these operators can be constructed systematically using the
bilinear 's as building blocks. We concentrate in particular on the
trilinear case and study the possible integrability of equations with one
dependent variable. The 5th order equation of the Lax-hierarchy as well as
Satsuma's lowest-order gauge invariant equation are shown to have simple
trilinear expressions. The formalism can be extended to an arbitrary degree of
multilinearity.Comment: 9 pages in plain Te
Relaxation of twisted vortices in the Faddeev-Skyrme model
We study vortex knotting in the Faddeev-Skyrme model. Starting with a
straight vortex line twisted around its axis we follow its evolution under
dissipative energy minimization dynamics. With low twist per unit length the
vortex forms a helical coil, but with higher twist numbers the vortex becomes
knotted or a ring is formed around the vortex.Comment: 7 pages, 8 jpg figure
On the parametrization of solutions of the Yang--Baxter equations
We study all five-, six-, and one eight-vertex type two-state solutions of
the Yang-Baxter equations in the form , and analyze the interplay of the `gauge' and `inversion' symmetries of
these solution. Starting with algebraic solutions, whose parameters have no
specific interpretation, and then using these symmetries we can construct a
parametrization where we can identify global, color and spectral parameters. We
show in particular how the distribution of these parameters may be changed by a
change of gauge.Comment: 19 pages in LaTe
Searching for integrable lattice maps using factorization
We analyze the factorization process for lattice maps, searching for
integrable cases. The maps were assumed to be at most quadratic in the
dependent variables, and we required minimal factorization (one linear factor)
after 2 steps of iteration. The results were then classified using algebraic
entropy. Some new models with polynomial growth (strongly associated with
integrability) were found. One of them is a nonsymmetric generalization of the
homogeneous quadratic maps associated with KdV (modified and Schwarzian), for
this new model we have also verified the "consistency around a cube".Comment: To appear in Journal of Physics A. Some changes in reference
Weak Lax pairs for lattice equations
We consider various 2D lattice equations and their integrability, from the
point of view of 3D consistency, Lax pairs and B\"acklund transformations. We
show that these concepts, which are associated with integrability, are not
strictly equivalent. In the course of our analysis, we introduce a number of
black and white lattice models, as well as variants of the functional
Yang-Baxter equation
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