1,336 research outputs found
On an integrable discretization of the modified Korteweg-de Vries equation
We find time discretizations for the two ''second flows'' of the
Ablowitz-Ladik hierachy. These discretizations are described by local equations
of motion, as opposed to the previously known ones, due to Taha and Ablowitz.
Certain superpositions of our maps allow a one-field reduction and serve
therefore as valid space-time discretizations of the modified Korteweg-de Vries
equation. We expect the performance of these discretizations to be much better
then that of the Taha-Ablowitz scheme. The way of finding interpolating
Hamiltonians for our maps is also indicated, as well as the solution of an
initial value problem in terms of matrix factorizations.Comment: 23 pages, LaTe
New integrable systems related to the relativistic Toda lattice
New integrable lattice systems are introduced, their different integrable
discretization are obtained. B\"acklund transformations between these new
systems and the relativistic Toda lattice (in the both continuous and discrete
time formulations) are established.Comment: LaTeX, 22 pp. Substantially extended version: several new systems
added
Why are the rational and hyperbolic Ruijsenaars-Schneider hierarchies governed by the same R-operators as the Calogero-Moser ones?
We demonstrate that in a certain gauge the Lax matrices of the rational and
hyperbolic Ruijsenaars--Schneider models have a quadratic -matrix Poisson
bracket which is an exact quadratization of the linear --matrix Poisson
bracket of the Calogero--Moser models. This phenomenon is explained by a
geometric derivation of Lax equations for arbitrary flows of both hierarchies,
which turn out to be governed by the same dynamical --operator.Comment: LaTeX, 18pp, a revised versio
Legendre transformations on the triangular lattice
The main purpose of the paper is to demonstrate that condition of invariance
with respect to the Legendre transformations allows effectively isolate the
class of integrable difference equations on the triangular lattice, which can
be considered as discrete analogues of relativistic Toda type lattices. Some of
obtained equations are new, up to the author knowledge. As an example, one of
them is studied in more details, in particular, its higher continuous
symmetries and zero curvature representation are found.Comment: 13 pages, late
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