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

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

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 $s=\pm 2$ (gravitational waves), $s=\pm 1$ (electromagnetic waves) and $s=\pm 1/2$ (neutrinos) is an Heun equation in disguise.Comment: 27 pages, corrected typo in eq. (1

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

Spherically Symmetric solutions in Multidimensional Gravity with the SU(2) Gauge Group as the Extra Dimensions

The multidimensional gravity on the principal bundle with the SU(2) gauge group is considered. The numerical investigation of the spherically symmetric metrics with the center of symmetry is made. The solution of the gravitational equations depends on the boundary conditions of the ``SU(2) gauge potential'' (off-diagonal metric components) at the symmetry center and on the type of symmetry (symmetrical or antisymmetrical) of these potentials. In the chosen range of the boundary conditions it is shown that there are two types of solutions: wormhole-like and flux tube. The physical application of such kind of solutions as quantum handles in a spacetime foam is discussed.Comment: misprints are correcte

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

Possible way out of the Hawking paradox: Erasing the information at the horizon

We show that small deviations from spherical symmetry, described by means of exact solutions to Einstein equations, provide a mechanism to "bleach" the information about the collapsing body as it falls through the aparent horizon, thereby resolving the information loss paradox. The resulting picture and its implication related to the Landauer's principle in the presence of a gravitational field, is discussed.Comment: 11 pages, Latex. Some comments added to answer to some raised questions. Typos corected. Final version, to appear in Int. J. Modern. Phys.