Symmetries of differential-difference dynamical systems in a two-dimensional lattice

Classification of differential-difference equation of the form $\ddot{u}_{nm}=F_{nm}\big(t, \{u_{pq}\}|_{(p,q)\in \Gamma}\big)$ are considered according to their Lie point symmetry groups. The set $\Gamma$ represents the point $(n,m)$ and its six nearest neighbors in a two-dimensional triangular lattice. It is shown that the symmetry group can be at most 12-dimensional for abelian symmetry algebras and 13-dimensional for nonsolvable symmetry algebras.Comment: 24 pages, 1 figur

Photo-excited zero-resistance states in the GaAs/AlGaAs system

The microwave-excited high mobility two-dimensional electron system exhibits, at liquid helium temperatures, vanishing resistance in the vicinity of $B = [4/(4j+1)] B_{f}$, where $B_{f} = 2\pi\textit{f}m^{*}/e$, m$^{*}$ is an effective mass, e is the charge, and \textit{f} is the microwave frequency. Here, we summarize some experimental results.Comment: 7 color figures, 5 page

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

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

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

Electronic Raman scattering of Tl-2223 and the symmetry of the supercon- ducting gap

Single crystalline Tl2Ba2Ca2Cu3O10 was studied using electronic Raman scattering. The renormalization of the scattering continuum was investigated as a function of the scattering geometry to determine the superconducting energy gap 2Delta(k). The A1g- and B2g-symmetry component show a linear frequency behaviour of the scattering intensity with a peak related to the energy gap, while the B1g-symmetry component shows a characteristic behaviour at higher frequencies. The observed frequency dependencies are consistent with a dx^2-y^2-wave symmetry of the gap and yield a ratio of 2Delta/k_BT_c=7.4. With the polarization of the scattered and incident light either parallel or perpendicular to the CuO2-planes a strong anisotropy due to the layered structure was detected, which indicates an almost 2 dimensional behaviour of this system.Comment: 2 pages, Postscript-file including 2 figures. Accepted for publication in the Proceedings of the M^2SHTSC IV Conference, Grenoble (France), 5-9 July 1994. Proceedings to be published in Physica C. Contact address: [email protected]

Kinetic Antiferromagnetism in the Triangular Lattice

We show that the motion of a single hole in the infinite $U$ Hubbard model with frustrated hopping leads to weak metallic antiferromagnetism of kinetic origin. An intimate relationship is demonstrated between the simplest versions of this problem in 1 and 2 dimensions, and two of the most subtle many body problems, namely the Heisenberg Bethe ring in 1-d and the 2-dimensional triangular lattice Heisenberg antiferromagnet.Comment: 10 pages, 2 figures, 5 supplementary figures; Figures fixe

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

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