11,105 research outputs found
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
The lattice Schwarzian KdV equation and its symmetries
In this paper we present a set of results on the symmetries of the lattice
Schwarzian Korteweg-de Vries (lSKdV) equation. We construct the Lie point
symmetries and, using its associated spectral problem, an infinite sequence of
generalized symmetries and master symmetries. We finally show that we can use
master symmetries of the lSKdV equation to construct non-autonomous
non-integrable generalized symmetries.Comment: 11 pages, no figures. Submitted to Jour. Phys. A, Special Issue SIDE
VI
Heun equation, Teukolsky equation, and type-D metrics
Starting with the whole class of type-D vacuum backgrounds with cosmological
constant we show that the separated Teukolsky equation for zero rest-mass
fields with spin (gravitational waves), (electromagnetic
waves) and (neutrinos) is an Heun equation in disguise.Comment: 27 pages, corrected typo in eq. (1
Cylindrically symmetric, static strings with a cosmological constant in Brans-Dicke theory
The static, cylindrically symmetric vacuum solutions with a cosmological
constant in the framework of the Brans-Dicke theory are investigated. Some of
these solutions admitting Lorentz boost invariance along the symmetry axis
correspond to local, straight cosmic strings with a cosmological constant. Some
physical properties of such solutions are studied. These strings apply
attractive or repulsive forces on the test particles. A smooth matching is also
performed with a recently introduced interior thick string solution with a
cosmological constant.Comment: 8 pages, Revtex; Published versio
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
Lie point symmetries of difference equations and lattices
A method is presented for finding the Lie point symmetry transformations
acting simultaneously on difference equations and lattices, while leaving the
solution set of the corresponding difference scheme invariant. The method is
applied to several examples. The found symmetry groups are used to obtain
particular solutions of differential-difference equations
Classification of discrete systems on a square lattice
We consider the classification up to a Möbius transformation of real linearizable and integrable partial difference equations with dispersion defined on a square lattice by the multiscale reduction around their harmonic solution. We show that the A1, A2, and A3 linearizability and integrability conditions constrain the number of parameters in the equation, but these conditions are insufficient for a complete characterization of the subclass of multilinear equations on a square lattice
Infinitely many symmetries and conservation laws for quad-graph equations via the Gardner method
The application of the Gardner method for generation of conservation laws to
all the ABS equations is considered. It is shown that all the necessary
information for the application of the Gardner method, namely B\"acklund
transformations and initial conservation laws, follow from the multidimensional
consistency of ABS equations. We also apply the Gardner method to an asymmetric
equation which is not included in the ABS classification. An analog of the
Gardner method for generation of symmetries is developed and applied to
discrete KdV. It can also be applied to all the other ABS equations
Vortex configurations of bosons in an optical lattice
The single vortex problem in a strongly correlated bosonic system is
investigated self-consistently within the mean-field theory of the Bose-Hubbard
model. Near the superfluid-Mott transition, the vortex core has a tendency
toward the Mott-insulating phase, with the core particle density approaching
the nearest commensurate value. If the nearest neighbor repulsion exists, the
charge density wave order may develop locally in the core. The evolution of the
vortex configuration from the strong to weak coupling regions is studied. This
phenomenon can be observed in systems of rotating ultra-cold atoms in optical
lattices and Josephson junction arraysComment: 4 pages, 4 figs, Accepted by Physics Review
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