Involutivity of integrals for sine-Gordon, modified KdV and potential KdV maps

Closed form expressions in terms of multi-sums of products have been given in \cite{Tranclosedform, KRQ} of integrals of sine-Gordon, modified Korteweg-de Vries and potential Korteweg-de Vries maps obtained as so-called $(p,-1)$-traveling wave reductions of the corresponding partial difference equations. We prove the involutivity of these integrals with respect to recently found symplectic structures for those maps. The proof is based on explicit formulae for the Poisson brackets between multi-sums of products.Comment: 24 page

The staircase method: integrals for periodic reductions of integrable lattice equations

We show, in full generality, that the staircase method provides integrals for mappings, and correspondences, obtained as traveling wave reductions of (systems of) integrable partial difference equations. We apply the staircase method to a variety of equations, including the Korteweg-De Vries equation, the five-point Bruschi-Calogero-Droghei equation, the QD-algorithm, and the Boussinesq system. We show that, in all these cases, if the staircase method provides r integrals for an n-dimensional mapping, with 2r<n, then one can introduce q<= 2r variables, which reduce the dimension of the mapping from n to q. These dimension-reducing variables are obtained as joint invariants of k-symmetries of the mappings. Our results support the idea that often the staircase method provides sufficiently many integrals for the periodic reductions of integrable lattice equations to be completely integrable. We also study reductions on other quad-graphs than the regular 2D lattice, and we prove linear growth of the multi-valuedness of iterates of high-dimensional correspondences obtained as reductions of the QD-algorithm.Comment: 40 pages, 23 Figure

Water depletion in the disk atmosphere of Herbig AeBe stars

We present high resolution (R = 100,000) L-band spectroscopy of 11 Herbig AeBe stars with circumstellar disks. The observations were obtained with the VLT/CRIRES to detect hot water and hydroxyl radical emission lines previously detected in disks around T Tauri stars. OH emission lines are detected towards 4 disks. The OH P4.5 (1+,1-) doublet is spectrally resolved as well as the velocity profile of each component of the doublet. Its characteristic double-peak profile demonstrates that the gas is in Keplerian rotation and points to an emitting region extending out to ~ 15-30 AU. The OH, emission correlates with disk geometry as it is mostly detected towards flaring disks. None of the Herbig stars analyzed here show evidence of hot water vapor at a sensitivity similar to that of the OH lines. The non-detection of hot water vapor emission indicates that the atmosphere of disks around Herbig AeBe stars are depleted of water molecules. Assuming LTE and optically thin emission we derive a lower limit to the OH/H2O column density ratio > 1 - 25 in contrast to T Tauri disks for which the column density ratio is 0.3 -- 0.4.Comment: Accepted for publication in Ap

Using discrete Darboux polynomials to detect and determine preserved measures and integrals of rational maps

In this Letter we propose a systematic approach for detecting and calculating preserved measures and integrals of a rational map. The approach is based on the use of cofactors and Discrete Darboux Polynomials and relies on the use of symbolic algebra tools. Given sufficient computing power, all rational preserved integrals can be found. We show, in two examples, how to use this method to detect and determine preserved measures and integrals of the considered rational maps.Comment: 8 pages, 1 Figur