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

Isotope Spectroscopy

The measurement of isotopic ratios provides a privileged insight both into nucleosynthesis and into the mechanisms operating in stellar envelopes, such as gravitational settling. In this article, we give a few examples of how isotopic ratios can be determined from high-resolution, high-quality stellar spectra. We consider examples of the lightest elements, H and He, for which the isotopic shifts are very large and easily measurable, and examples of heavier elements for which the determination of isotopic ratios is more difficult. The presence of 6Li in the stellar atmospheres causes a subtle extra depression in the red wing of the 7Li 670.7 nm doublet which can only be detected in spectra of the highest quality. But even with the best spectra, the derived $^6$Li abundance can only be as good as the synthetic spectra used for their interpretation. It is now known that 3D non-LTE modelling of the lithium spectral line profiles is necessary to account properly for the intrinsic line asymmetry, which is produced by convective flows in the atmospheres of cool stars, and can mimic the presence of 6Li. We also discuss briefly the case of the carbon isotopic ratio in metal-poor stars, and provide a new determination of the nickel isotopic ratios in the solar atmosphere.Comment: AIP Thinkshop 10 "High resolution optical spectroscopy", invited talk, AN in pres

The Solar Photospheric Nitrogen Abundance: Determination with 3D and 1D Model Atmospheres

We present a new determination of the solar nitrogen abundance making use of 3D hydrodynamical modelling of the solar photosphere, which is more physically motivated than traditional static 1D models. We selected suitable atomic spectral lines, relying on equivalent width measurements already existing in the literature. For atmospheric modelling we used the co 5 bold 3D radiation hydrodynamics code. We investigated the influence of both deviations from local thermodynamic equilibrium (non-LTE effects) and photospheric inhomogeneities (granulation effects) on the resulting abundance. We also compared several atlases of solar flux and centre-disc intensity presently available. As a result of our analysis, the photospheric solar nitrogen abundance is A(N) = 7.86 +/- 0.12.Comment: 6 pages, 4 figure

Higher analogues of the discrete-time Toda equation and the quotient-difference algorithm

The discrete-time Toda equation arises as a universal equation for the relevant Hankel determinants associated with one-variable orthogonal polynomials through the mechanism of adjacency, which amounts to the inclusion of shifted weight functions in the orthogonality condition. In this paper we extend this mechanism to a new class of two-variable orthogonal polynomials where the variables are related via an elliptic curve. This leads to a `Higher order Analogue of the Discrete-time Toda' (HADT) equation for the associated Hankel determinants, together with its Lax pair, which is derived from the relevant recurrence relations for the orthogonal polynomials. In a similar way as the quotient-difference (QD) algorithm is related to the discrete-time Toda equation, a novel quotient-quotient-difference (QQD) scheme is presented for the HADT equation. We show that for both the HADT equation and the QQD scheme, there exists well-posed $s$-periodic initial value problems, for almost all \s\in\Z^2. From the Lax-pairs we furthermore derive invariants for corresponding reductions to dynamical mappings for some explicit examples.Comment: 38 page

Integrable and superintegrable systems associated with multi-sums of products

We construct and study certain Liouville integrable, superintegrable, and non-commutative integrable systems, which are associated with multi-sums of products.Comment: 26 pages, submitted to Proceedings of the royal society