Linear Asymptotic Stability and Modulation Behavior near Periodic Waves of the Korteweg-de Vries Equation

We provide a detailed study of the dynamics obtained by linearizing the Korteweg-de Vries equation about one of its periodic traveling waves, a cnoidal wave. In a suitable sense, linearly analogous to space-modulated stability, we prove global-in-time bounded stability in any Sobolev space, and asymptotic stability of dispersive type. Furthermore, we provide both a leading-order description of the dynamics in terms of slow modulation of local parameters and asymptotic modulation systems and effective initial data for the evolution of those parameters. This requires a global-in-time study of the dynamics generated by a non normal operator with non constant coefficients. On the road we also prove estimates on oscillatory integrals particularly suitable to derive large-time asymptotic systems that could be of some general interest

Space-modulated Stability and Averaged Dynamics

In this brief note we give a brief overview of the comprehensive theory, recently obtained by the author jointly with Johnson, Noble and Zumbrun, that describes the nonlinear dynamics about spectrally stable periodic waves of parabolic systems and announce parallel results for the linearized dynamics near cnoidal waves of the Korteweg-de Vries equation. The latter are expected to contribute to the development of a dispersive theory, still to come.Comment: Proceedings of the "Journ\'ees \'Equations aux d\'eriv\'ees partielles", Roscoff 201

Using presence-absence data to establish reserve selection procedures that are robust to temporal species turnover

Previous studies suggest that a network of nature reserves with maximum efficiency (obtained by selecting the minimum area such that each species is represented once) is likely to be insufficient to maintain species in the network over time. Here, we test the performance of three selection strategies which require presence-absence data, two of them previously proposed (multiple representations and selecting an increasing percentage of each species' range) and a novel one based on selecting the site where each species has exhibited a higher permanence rate in the past. Multiple representations appear to be a safer strategy than selecting a percentage of range because the former gives priority to rarer species while the latter favours the most widespread. The most effective strategy was the one based on the permanence rate, indicating that the robustness of reserve networks can be improved by adopting reserve selection procedures that integrate information about the relative value of sites. This strategy was also very efficient, suggesting that the investment made in the monitoring schemes may be compensated for by a lower cost in reserve acquisition

Generalized Teleparallel Theory

We construct a theory in which the gravitational interaction is described only by torsion, but that generalizes the Teleparallel Theory still keeping the invariance of local Lorentz transformations in one particular case. We show that our theory falls, to a certain limit of a real parameter, in the $f(\bar{R})$ Gravity or, to another limit of the same real parameter, in a modified $f(T)$ Gravity, interpolating between these two theories and still can fall on several other theories. We explicitly show the equivalence with $f(\bar{R})$ Gravity for cases of Friedmann-Lemaitre-Robertson-Walker flat metric for diagonal tetrads, and a metric with spherical symmetry for diagonal and non-diagonal tetrads. We do still four applications, one in the reconstruction of the de Sitter universe cosmological model, for obtaining a static spherically symmetric solution type-de Sitter for a perfect fluid, for evolution of the state parameter $\omega_{DE}$ and for the thermodynamics to the apparent horizon.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1503.07427, arXiv:1503.0785