On the Nonlocal Equations and Nonlocal Charges Associated with the Harry Dym Hierarchy

A large class of nonlocal equations and nonlocal charges for the Harry Dym hierarchy is exhibited. They are obtained from nonlocal Casimirs associated with its bi-Hamiltonian structure. The Lax representation for some of these equations is also given.Comment: to appear in Journal of Mathematical Physics, 17 pages, Late

Deformed Harry Dym and Hunter-Zheng Equations

We study the deformed Harry Dym and Hunter-Zheng equations with two arbitrary deformation parameters. These reduce to various other known models in appropriate limits. We show that both these systems are bi-Hamiltonian with the same Hamiltonian structures. They are integrable and belong to the same hierarchy corresponding to positive and negative flows. We present the Lax pair description for both the systems and construct the conserved charges of negative order from the Lax operator. For the deformed Harry Dym equation, we construct the non-standard Lax representation for two special classes of values of the deformation parameters. In general, we argue that a non-standard description will involve a pseudo-differential operator of infinite order.Comment: Latex file, 15 page

²¹⁰Pb- ²²⁶Ra chronology reveals rapid growth rate of Madrepora oculata and Lophelia pertusa on world's largest cold-water coral reef

Here we show the use of the 210Pb- 226Ra excess method to determine the growth rate of two corals from the world's largest known cold-water coral reef, Røst Reef, north of the Arctic circle off Norway. Colonies of each of the two species that build the reef, Lophelia pertusa and Madrepora oculata, were collected alive at 350 m depth using a submersible. Pb and Ra isotopes were measured along the major growth axis of both specimens using low level alpha and gamma spectrometry and trace element compositions were studied. 210Pb and 226Ra differ in the way they are incorporated into coral skeletons. Hence, to assess growth rates, we considered the exponential decrease of initially incorporated 210Pb, as well as the increase in 210Pb from the decay of 226Ra and contamination with 210Pb associated with Mn-Fe coatings that we were unable to remove completely from the oldest parts of the skeletons. 226Ra activity was similar in both coral species, so, assuming constant uptake of 210Pb through time, we used the 210Pb- 226Ra chronology to calculate growth rates. The 45.5 cm long branch of M. oculata was 31 yr with an average linear growth rate of 14.4 ± 1.1 mm yr -1 (2.6 polyps per year). Despite cleaning, a correction for Mn-Fe oxide contamination was required for the oldest part of the colony; this correction corroborated our radiocarbon date of 40 yr and a mean growth rate of 2 polyps yr -1. This rate is similar to the one obtained in aquarium experiments under optimal growth conditions. For the 80 cm-long L. pertusa colony, metal-oxide contamination remained in both the middle and basal part of the coral skeleton despite cleaning, inhibiting similar age and growth rate estimates. The youngest part of the colony was free of metal oxides and this 15 cm section had an estimated a growth rate of 8 mm yr -1, with high uncertainty (∼1 polyp every two to three years). We are less certain of this 210Pb growth rate estimate which is within the lowermost ranges of previous growth rate estimates. We show that 210Pb- 226Ra dating can be successfully applied to determine the age and growth rate of framework-forming cold-water corals if Mn-Fe oxide deposits can be removed. Where metal oxides can be removed, large M. oculata and L. pertusa skeletons provide archives for studies of intermediate water masses with an up to annual time resolution and spanning over many decades. © 2012 Author(s)

Electron Wave Filters from Inverse Scattering Theory

Semiconductor heterostructures with prescribed energy dependence of the transmittance can be designed by combining: {\em a)} Pad\'e approximant reconstruction of the S-matrix; {\em b)} inverse scattering theory for Schro\"dinger's equation; {\em c)} a unitary transformation which takes into account the variable mass effects. The resultant continuous concentration profile can be digitized into an easily realizable rectangular-wells structure. For illustration, we give the specifications of a 2 narrow band-pass 12 layer $Al_cGa_{1-c}As$ filter with the high energy peak more than {\em twice narrower} than the other.Comment: 4 pages, Revtex with one eps figur

Nonlinear Evolution Equations Invariant Under Schroedinger Group in three-dimensional Space-time

A classification of all possible realizations of the Galilei, Galilei-similitude and Schroedinger Lie algebras in three-dimensional space-time in terms of vector fields under the action of the group of local diffeomorphisms of the space \R^3\times\C is presented. Using this result a variety of general second order evolution equations invariant under the corresponding groups are constructed and their physical significance are discussed

Use of specific Green's functions for solving direct problems involving a heterogeneous rigid frame porous medium slab solicited by acoustic waves

A domain integral method employing a specific Green's function (i.e., incorporating some features of the global problem of wave propagation in an inhomogeneous medium) is developed for solving direct and inverse scattering problems relative to slab-like macroscopically inhomogeneous porous obstacles. It is shown how to numerically solve such problems, involving both spatially-varying density and compressibility, by means of an iterative scheme initialized with a Born approximation. A numerical solution is obtained for a canonical problem involving a two-layer slab.Comment: submitted to Math.Meth.Appl.Sc

Action versus Result-Oriented Schemes in a Grassland Agroecosystem: A Dynamic Modelling Approach

Effects of agri-environment schemes (AES) on biodiversity remain controversial. While most AES are action-oriented, result-oriented and habitat-oriented schemes have recently been proposed as a solution to improve AES efficiency. The objective of this study was to compare action-oriented, habitat-oriented and result-oriented schemes in terms of ecological and productive performance as well as in terms of management flexibility. We developed a dynamic modelling approach based on the viable control framework to carry out a long term assessment of the three schemes in a grassland agroecosystem. The model explicitly links grazed grassland dynamics to bird population dynamics. It is applied to lapwing conservation in wet grasslands in France. We ran the model to assess the three AES scenarios. The model revealed the grazing strategies respecting ecological and productive constraints specific to each scheme. Grazing strategies were assessed by both their ecological and productive performance. The viable control approach made it possible to obtain the whole set of viable grazing strategies and therefore to quantify the management flexibility of the grassland agroecosystem. Our results showed that habitat and result-oriented scenarios led to much higher ecological performance than the action-oriented one. Differences in both ecological and productive performance between the habitat and result-oriented scenarios were limited. Flexibility of the grassland agroecosystem in the result-oriented scenario was much higher than in that of habitat-oriented scenario. Our model confirms the higher flexibility as well as the better ecological and productive performance of result-oriented schemes. A larger use of result-oriented schemes in conservation may also allow farmers to adapt their management to local conditions and to climatic variations

Symmetry, Local Linearization, and Gauge Classification of the Doebner-Goldin Equation

For the family of nonlinear Schr\"odinger equations derived by H.-D.~Doebner and G.A.~Goldin (J.Phys.A 27, 1771) we calculate the complete set of Lie symmetries. For various subfamilies we find different finite and infinite dimensional Lie symmetry algebras. Two of the latter lead to a local transformation linearizing the particular subfamily. One type of these transformations leaves the whole family of equations invariant, giving rise to a gauge classification of the family. The Lie symmetry algebras and their corresponding subalgebras are finally characterized by gauge invariant parameters.Comment: 17 pages, LaTeX, 1 figure, to appear in Reports on Mathematical Physic