11,018 research outputs found
Lie discrete symmetries of lattice equations
We extend two of the methods previously introduced to find discrete
symmetries of differential equations to the case of difference and
differential-difference equations. As an example of the application of the
methods, we construct the discrete symmetries of the discrete Painlev\'e I
equation and of the Toda lattice equation
Multiscale expansion and integrability properties of the lattice potential KdV equation
We apply the discrete multiscale expansion to the Lax pair and to the first
few symmetries of the lattice potential Korteweg-de Vries equation. From these
calculations we show that, like the lowest order secularity conditions give a
nonlinear Schroedinger equation, the Lax pair gives at the same order the
Zakharov and Shabat spectral problem and the symmetries the hierarchy of point
and generalized symmetries of the nonlinear Schroedinger equation.Comment: 10 pages, contribution to the proceedings of the NEEDS 2007
Conferenc
Mesons and Flavor on the Conifold
We explore the addition of fundamental matter to the Klebanov-Witten field
theory. We add probe D7-branes to the theory obtained from placing
D3-branes at the tip of the conifold and compute the meson spectrum for the
scalar mesons. In the UV limit of massless quarks we find the exact dimensions
of the associated operators, which exhibit a simple scaling in the large-charge
limit. For the case of massive quarks we compute the spectrum of scalar mesons
numerically.Comment: 19 pages, 3 figures, v2: typos fixe
Discrete derivatives and symmetries of difference equations
We show on the example of the discrete heat equation that for any given
discrete derivative we can construct a nontrivial Leibniz rule suitable to find
the symmetries of discrete equations. In this way we obtain a symmetry Lie
algebra, defined in terms of shift operators, isomorphic to that of the
continuous heat equation.Comment: submitted to J.Phys. A 10 Latex page
Rotating saddle trap as Foucault's pendulum
One of the many surprising results found in the mechanics of rotating systems
is the stabilization of a particle in a rapidly rotating planar saddle
potential. Besides the counterintuitive stabilization, an unexpected
precessional motion is observed. In this note we show that this precession is
due to a Coriolis-like force caused by the rotation of the potential. To our
knowledge this is the first example where such force arises in an inertial
reference frame. We also propose an idea of a simple mechanical demonstration
of this effect.Comment: 13 pages, 9 figure
Multiscale reduction of discrete nonlinear Schroedinger equations
We use a discrete multiscale analysis to study the asymptotic integrability
of differential-difference equations. In particular, we show that multiscale
perturbation techniques provide an analytic tool to derive necessary
integrability conditions for two well-known discretizations of the nonlinear
Schroedinger equation.Comment: 12 page
Continuous Symmetries of Difference Equations
Lie group theory was originally created more than 100 years ago as a tool for
solving ordinary and partial differential equations. In this article we review
the results of a much more recent program: the use of Lie groups to study
difference equations. We show that the mismatch between continuous symmetries
and discrete equations can be resolved in at least two manners. One is to use
generalized symmetries acting on solutions of difference equations, but leaving
the lattice invariant. The other is to restrict to point symmetries, but to
allow them to also transform the lattice.Comment: Review articl
Stationary structures in two-dimensional continuous Heisenberg ferromagnetic spin system
Stationary structures in a classical isotropic two-dimensional continuous
Heisenberg ferromagnetic spin system are studied in the framework of the
(2+1)-dimensional Landau-Lifshitz model. It is established that in the case of
\vec S (\vec r, t)= \vec S (\vec r - \vec v t) the Landau-Lifshitz equation is
closely related to the Ablowitz-Ladik hierarchy. This relation is used to
obtain soliton structures, which are shown to be caused by joint action of
nonlinearity and spatial dispersion, contrary to the well-known one-dimensional
solitons which exist due to competition of nonlinearity and temporal
dispersion. We also present elliptical quasiperiodic stationary solutions of
the stationary (2+1)-dimensional Landau-Lifshitz equation.Comment: Archive version is already official Published by JNMP at
http://www.sm.luth.se/math/JNMP
- …