Difference schemes with point symmetries and their numerical tests

Symmetry preserving difference schemes approximating second and third order ordinary differential equations are presented. They have the same three or four-dimensional symmetry groups as the original differential equations. The new difference schemes are tested as numerical methods. The obtained numerical solutions are shown to be much more accurate than those obtained by standard methods without an increase in cost. For an example involving a solution with a singularity in the integration region the symmetry preserving scheme, contrary to standard ones, provides solutions valid beyond the singular point.Comment: 26 pages 7 figure

The Taming of QCD by Fortran 90

We implement lattice QCD using the Fortran 90 language. We have designed machine independent modules that define fields (gauge, fermions, scalars, etc...) and have defined overloaded operators for all possible operations between fields, matrices and numbers. With these modules it is very simple to write QCD programs. We have also created a useful compression standard for storing the lattice configurations, a parallel implementation of the random generators, an assignment that does not require temporaries, and a machine independent precision definition. We have tested our program on parallel and single processor supercomputers obtaining excellent performances.Comment: Talk presented at LATTICE96 (algorithms) 3 pages, no figures, LATEX file with ESPCRC2 style. More information available at: http://hep.bu.edu/~leviar/qcdf90.htm

A new two-dimensional lattice model that is "consistent around a cube"

For two-dimensional lattice equations one definition of integrability is that the model can be naturally and consistently extended to three dimensions, i.e., that it is "consistent around a cube" (CAC). As a consequence of CAC one can construct a Lax pair for the model. Recently Adler, Bobenko and Suris conducted a search based on this principle and certain additional assumptions. One of those assumptions was the "tetrahedron property", which is satisfied by most known equations. We present here one lattice equation that satisfies the consistency condition but does not have the tetrahedron property. Its Lax pair is also presented and some basic properties discussed.Comment: 8 pages in LaTe

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

On the Integrability of the Discrete Nonlinear Schroedinger Equation

In this letter we present an analytic evidence of the non-integrability of the discrete nonlinear Schroedinger equation, a well-known discrete evolution equation which has been obtained in various contexts of physics and biology. We use a reductive perturbation technique to show an obstruction to its integrability.Comment: 4 pages, accepted in EP

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

Discrete Reductive Perturbation Technique

We expand a partial difference equation (P$\Delta$E) on multiple lattices and obtain the P$\Delta$E which governs its far field behaviour. The perturbative--reductive approach is here performed on well known nonlinear P$\Delta$Es, both integrable and non integrable. We study the cases of the lattice modified Korteweg--de Vries (mKdV) equation, the Hietarinta equation, the lattice Volterra--Kac--Van Moerbeke (VKVM) equation and a non integrable lattice KdV equation. Such reductions allow us to obtain many new P$\Delta$Es of the nonlinear Schr\"odinger (NLS) type.Comment: 18 pages, 1 figure. submitted to Journal of Mathematical Physic

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

Magnetostrictive Neel ordering of the spin-5/2 ladder compound BaMn2O3: distortion-induced lifting of geometrical frustration

The crystal structure and the magnetism of BaMn$_2$O$_3$ have been studied by thermodynamic and by diffraction techniques using large single crystals and powders. BaMn$_2$O$_3$ is a realization of a $S = 5/2$ spin ladder as the magnetic interaction is dominant along 180$^\circ$ Mn-O-Mn bonds forming the legs and the rungs of a ladder. The temperature dependence of the magnetic susceptibility exhibits well-defined maxima for all directions proving the low-dimensional magnetic character in BaMn$_2$O$_3$. The susceptibility and powder neutron diffraction data, however, show that BaMn$_2$O$_3$ exhibits a transition to antiferromagnetic order at 184 K, in spite of a full frustration of the nearest-neighbor inter-ladder coupling in the orthorhombic high-temperature phase. This frustration is lifted by a remarkably strong monoclinic distortion which accompanies the magnetic transition.Comment: 9 pages, 8 figures, 2 tables; in V1 fig. 2 was included twice and fig. 4 was missing; this has been corrected in V

Integrability of Differential-Difference Equations with Discrete Kinks

In this article we discuss a series of models introduced by Barashenkov, Oxtoby and Pelinovsky to describe some discrete approximations to the \phi^4 theory which preserve travelling kink solutions. We show, by applying the multiple scale test that they have some integrability properties as they pass the A_1 and A_2 conditions. However they are not integrable as they fail the A_3 conditions.Comment: submitted to the Proceedings of the workshop "Nonlinear Physics: Theory and Experiment.VI" in a special issue di Theoretical and Mathematical Physic