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

Invariant differential equations and the Adler-Gel'fand-Dikii bracket

In this paper we find an explicit formula for the most general vector evolution of curves on RP^{n-1} invariant under the projective action of SL(n,R). When this formula is applied to the projectivization of solution curves of scalar Lax operators with periodic coefficients, one obtains a corresponding evolution in the space of such operators. We conjecture that this evolution is identical to the second KdV Hamiltonian evolution under appropriate conditions. These conditions give a Hamiltonian interpretation of general vector differential invariants for the projective action of SL(n,R), namely, the SL(n,R) invariant evolution can be written so that a general vector differential invariant corresponds to the Hamiltonian pseudo-differential operator. We find common coordinates and simplify both evolutions so that one can attempt to prove the equivalence for arbitrary n

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

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

Dressing chain for the acoustic spectral problem

The iterations are studied of the Darboux transformation for the generalized Schroedinger operator. The applications to the Dym and Camassa-Holm equations are considered.Comment: 16 pages, 6 eps figure

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 Multiscale Analysis: A Biatomic Lattice System

We discuss a discrete approach to the multiscale reductive perturbative method and apply it to a biatomic chain with a nonlinear interaction between the atoms. This system is important to describe the time evolution of localized solitonic excitations. We require that also the reduced equation be discrete. To do so coherently we need to discretize the time variable to be able to get asymptotic discrete waves and carry out a discrete multiscale expansion around them. Our resulting nonlinear equation will be a kind of discrete Nonlinear Schr\"odinger equation. If we make its continuum limit, we obtain the standard Nonlinear Schr\"odinger differential equation

Point Symmetries of Generalized Toda Field Theories

A class of two-dimensional field theories with exponential interactions is introduced. The interaction depends on two ``coupling'' matrices and is sufficiently general to include all Toda field theories existing in the literature. Lie point symmetries of these theories are found for an infinite, semi-infinite and finite number of fields. Special attention is accorded to conformal invariance and its breaking.Comment: 25 pages, no figures, Latex fil

The Imaging Magnetograph eXperiment (IMaX) for the Sunrise balloon-borne solar observatory

The Imaging Magnetograph eXperiment (IMaX) is a spectropolarimeter built by four institutions in Spain that flew on board the Sunrise balloon-borne telesocope in June 2009 for almost six days over the Arctic Circle. As a polarimeter IMaX uses fast polarization modulation (based on the use of two liquid crystal retarders), real-time image accumulation, and dual beam polarimetry to reach polarization sensitivities of 0.1%. As a spectrograph, the instrument uses a LiNbO3 etalon in double pass and a narrow band pre-filter to achieve a spectral resolution of 85 mAA. IMaX uses the high Zeeman sensitive line of Fe I at 5250.2 AA and observes all four Stokes parameters at various points inside the spectral line. This allows vector magnetograms, Dopplergrams, and intensity frames to be produced that, after reconstruction, reach spatial resolutions in the 0.15-0.18 arcsec range over a 50x50 arcsec FOV. Time cadences vary between ten and 33 seconds, although the shortest one only includes longitudinal polarimetry. The spectral line is sampled in various ways depending on the applied observing mode, from just two points inside the line to 11 of them. All observing modes include one extra wavelength point in the nearby continuum. Gauss equivalent sensitivities are four Gauss for longitudinal fields and 80 Gauss for transverse fields per wavelength sample. The LOS velocities are estimated with statistical errors of the order of 5-40 m/s. The design, calibration and integration phases of the instrument, together with the implemented data reduction scheme are described in some detail.Comment: 17 figure