Group classification of (1+1)-Dimensional Schr\"odinger Equations with Potentials and Power Nonlinearities

We perform the complete group classification in the class of nonlinear Schr\"odinger equations of the form $i\psi_t+\psi_{xx}+|\psi|^\gamma\psi+V(t,x)\psi=0$ where $V$ is an arbitrary complex-valued potential depending on $t$ and $x,$ $\gamma$ is a real non-zero constant. We construct all the possible inequivalent potentials for which these equations have non-trivial Lie symmetries using a combination of algebraic and compatibility methods. The proposed approach can be applied to solving group classification problems for a number of important classes of differential equations arising in mathematical physics.Comment: 10 page

L'accumulation et l'élimination de cadmium par deux mousses aquatiques, Fontinalis dalecarlica et Platyphypnidium ripariodes : Influence de la concentration de Cd, du temps d'exposition, de la dureté de l'eau et de l'espèce de mousses

Cette étude en laboratoire traite de l'accumulation et de l'élimination du Cd réalisées par deux mousses aquatiques indigènes du Québec, Fontinalis dalecarlica et Platyhypnidium riparioides. Les expositions au Cd étaient de 0 (témoin), 0,8, 2 et 10 µg·L-1, concentrations retrouvées en milieu naturel (non contaminé) et contaminé. Les expériences ont été réalisées à trois niveaux de dureté de l'eau (10 à 15, 40 à 50, 80 à 100 mg·L-1 de CaCO3), à alcalinité constante (80 à 100 mg·L-1 de CaCO3) et à pH stable (7,30) durant une période de 28 jours. Les facteurs d'augmentation des concentrations (FAC) ont démontré une diminution de l'accumulation totale de Cd dans les mousses dans 75% des cas lorsque la dureté de l'eau passe de très douce à dure. Les facteurs de contamination résiduelle (FCR) démontrent la lenteur de l'élimination du Cd par les mousses, et ce, indépendamment de la dureté de l'eau ou de la contamination préalablement subie. Deux équations de régression multiple par étape (Stepwise) ont été établies pour expliquer les facteurs influençant l'accumulation et l'élimination de Cd réalisées par les mousses. Les variables indépendantes incluses dans les équations linéaires de prédiction pour l'accumulation et l'élimination étaient la concentration de Cd dans l'eau, le temps d'exposition, la dureté de l'eau, l'espèce de mousses utilisée et/ou les interactions de ces variables. Les équations linéaires de prédiction pour l'accumulation et l'élimination ont permis d'expliquer respectivement 92% et 71% de la variance observée. Cette identification des principaux facteurs influençant l'accumulation et l'élimination du Cd dans les mousses est d'une grande importance pour la compréhension des processus complexes dirigeant l'absortion des métaux lourds par des organismes vivants. Les équations permettent également de mieux expliquer les interactions engendrées par la variation de divers paramètres sur l'accumulation et l'élimination du Cd par les mousses aquatiques.Aquatic mosses have played a large part in the assessment of toxic elements in water. The advantage of mosses over direct water sampling is that the use of the former lessens spatial and temporal variations, enhances the level of contaminant identification by concentrating toxic elements, and provides information relative to the bioavailable portion. However, the concentration of metals that can be measured in mosses is not a reliable indicator of the concentration of toxic elements in the water, which is why we need to model the bioaccumulation phenomenon.The present laboratory study deals with the accumulation and elimination of Cd by two indigenous Quebec aquatic mosses: Fontinalis dalecarlica and Platyhypnidium riparioides. The previously acclimatized mosses were treated with different concentrations of Cd, three different levels of water hardness, a constant alkalinity and constant pH level for a period of 28 days, in order to establish their bioaccumulative capacity. Cd exposure concentrations were 0 (control), 0.8, 2 and 10 mg·L-1, with a replication at 10 mg·L-1. The experiments were carried out at three levels of water hardness (10 to 15, 40 to 50, 80 to 100 mg·L-1 of CaCO3), with a constant degree of alkalinity (80 to 100 mg·L-1 of CaCO3) and stable pH (7.30). The mosses subsequently went through an elimination period (Cd-free water) of 28 days. The triplicate moss samples were mineralized using nitric acid and all Cd measurements were made by atomic absorption spectrophotometry. The results indicate that the water chemistry conditions remained stable and were properly controlled. The aquatic mosses demonstrated a considerable ability to absorb and adsorb Cd: the measured Cd water concentrations were less than the nominal concentrations. Nonetheless, moss uptake of Cd improves with an increase in Cd contamination and the concentration factors (CF) range from 6 to 122. For the same exposure concentration, the CF drops in some 63% of those instances where water hardness rises from very soft, through soft, to hard. In 75% of the cases there is a drop in CF when water hardness increases directly from very soft to hard. With a stable concentration (e.g. 2 mg·L-1), F. dalecarlica has respective CFs of 26.3, 22.2 and 18, which demonstrates the negative gradation of Cd accumulation as water hardness increases. The residual contamination factors (RCF) bear witness to the slow rate of Cd elimination by the mosses, irrespective of the level of water hardness or of any previous contamination. The elimination factor for RCF is never greater than 2. Mosses take up metals faster than they can eliminate them and have a memory of past contaminations, which is an advantage when it comes to studying ad hoc and/or sporadic contamination phenomena.Two stepwise multiple regression equations have been set up to explain the factors that impact on accumulation and elimination of Cd by mosses. The variables included in the equations were: level of Cd concentration in the water; exposure time; water hardness; the moss species involved, and/or the interactions between these variables. The predictive linear equations for the accumulation and elimination provided explanations for 92% and 71% respectively of the observed variances. The predictive linear equation for accumulation establishes that the length of exposure is the principal parameter responsible for the uptake of Cd by the aquatic mosses. It shows that the accumulation of Cd by the mosses is initially influenced by the level of Cd concentration in the water, but also depends on the length of time over which the bryophytes are exposed to this concentration. Thus, the higher the Cd concentration, the shorter the exposure time for the moss contamination, and vice versa. The second variable is the effect of water hardness taken together with the exposure time. This is a negative variable: the greater the increase in water hardness, the greater the exposure time required to obtain the same degree of moss contamination. This is indicative of the impact of Ca++ and Mg++ on moss uptake. The impact of water hardness is probably the consequence of the availability of or preference of plant-binding sites for Ca++ and Mg++ ions, thus reducing the number of available locations for Cd accumulation. Water hardness and Cd concentration levels are the third variable in this equation and are probably linked to the effect of water hardness on the bioavailability of Cd for the mosses. This variable may also explain why the increase in Cd concentration levels in the water lessens the impact of water hardness on the total accumulation of Cd in the mosses. Finally, the equation identifies a greater level of accumulation in the P. riparoides.Release linear regression shows that the absence of Cd in the water is the major parameter in the elimination of Cd by aquatic mosses. We should remember that the bryophytes are seeking to achieve a steady state condition with their environment, since the Cd is an element that is neither regulated or essential. Its elimination has little to do with water hardness, but is caused by the inversion of a diffusion gradient when the environment is no longer Cd contaminated. During the elimination process, the Ca++ and Mg++ ions have no real impact on the release of Cd by the mosses. The length of prior exposure does affect elimination: the greater it is, the longer the release period required for moss decontamination. Exposure time is less important during elimination than during accumulation. Elimination is a very slow process, and a longer study would probably have shown that this is a major factor in the elimination of moss-accumulated Cd.The present identification of the major factors impacting on the accumulation and elimination of Cd in mosses is extremely important if we are to understand the complex processes that determine the absorption of heavy metals by living organisms. The equations also allow us to better explain the interactions caused by variations in the different parameters with respect to the accumulation and elimination of Cd by aquatic mosses

Exact Soliton-like Solutions of the Radial Gross-Pitaevskii Equation

We construct exact ring soliton-like solutions of the cylindrically symmetric (i.e., radial) Gross- Pitaevskii equation with a potential, using the similarity transformation method. Depending on the choice of the allowed free functions, the solutions can take the form of stationary dark or bright rings whose time dependence is in the phase dynamics only, or oscillating and bouncing solutions, related to the second Painlev\'e transcendent. In each case the potential can be chosen to be time-independent.Comment: 8 pages, 7 figures. Version 2: stability analysis of the dark solutio

Dynamics of Large-Scale Plastic Deformation and the Necking Instability in Amorphous Solids

We use the shear transformation zone (STZ) theory of dynamic plasticity to study the necking instability in a two-dimensional strip of amorphous solid. Our Eulerian description of large-scale deformation allows us to follow the instability far into the nonlinear regime. We find a strong rate dependence; the higher the applied strain rate, the further the strip extends before the onset of instability. The material hardens outside the necking region, but the description of plastic flow within the neck is distinctly different from that of conventional time-independent theories of plasticity.Comment: 4 pages, 3 figures (eps), revtex4, added references, changed and added content, resubmitted to PR

Symmetries of differential-difference dynamical systems in a two-dimensional lattice

Classification of differential-difference equation of the form $\ddot{u}_{nm}=F_{nm}\big(t, \{u_{pq}\}|_{(p,q)\in \Gamma}\big)$ are considered according to their Lie point symmetry groups. The set $\Gamma$ represents the point $(n,m)$ and its six nearest neighbors in a two-dimensional triangular lattice. It is shown that the symmetry group can be at most 12-dimensional for abelian symmetry algebras and 13-dimensional for nonsolvable symmetry algebras.Comment: 24 pages, 1 figur

Efficacy of Online Training for Improving Camp Staff Competency

Preparing competent staff is a critical issue within the camp community. This quasi-experimental study examined the effectiveness of an online course for improving staff competency in camp healthcare practices among college-aged camp staff and a comparison group (N = 55). We hypothesized that working in camp would increase competency test scores due to opportunities for staff to experientially apply knowledge learned online. Hierarchical linear modeling was used to analyse the cross-level effects of a between-individuals factor (assignment to experimental or comparison group) and within-individual effects of time (pre-test, post-test #1, and post-test #2) on online course test scores. At post-test #2, the difference in average test scores between groups was ~30 points, with the treatment group scoring lower on average than the comparison group. Factors that may have influenced these findings are explored, including fatigue and the limited durability of online learning. Recommendations for research and practice are discussed

New results on group classification of nonlinear diffusion-convection equations

Using a new method and additional (conditional and partial) equivalence transformations, we performed group classification in a class of variable coefficient $(1+1)$-dimensional nonlinear diffusion-convection equations of the general form $f(x)u_t=(D(u)u_x)_x+K(u)u_x.$ We obtain new interesting cases of such equations with the density $f$ localized in space, which have large invariance algebra. Exact solutions of these equations are constructed. We also consider the problem of investigation of the possible local trasformations for an arbitrary pair of equations from the class under consideration, i.e. of describing all the possible partial equivalence transformations in this class.Comment: LaTeX2e, 19 page

Helmholtz bright and boundary solitons

We report, for the first time, exact analytical boundary solitons of a generalized cubic-quintic Non-Linear Helmholtz (NLH) equation. These solutions have a linked-plateau topology that is distinct from conventional dark soliton solutions; their amplitude and intensity distributions are spatially delocalized and connect regions of finite and zero wave-field disturbances (suggesting also the classification as 'edge solitons'). Extensive numerical simulations compare the stability properties of recently-reported Helmholtz bright solitons, for this type of polynomial non-linearity, to those of the new boundary solitons. The latter are found to possess a remarkable stability characteristic, exhibiting robustness against perturbations that would otherwise lead to the destabilizing of their bright-soliton counterpart

Modulational instability of bright solitary waves in incoherently coupled nonlinear Schr\"odinger equations

We present a detailed analysis of the modulational instability (MI) of ground-state bright solitary solutions of two incoherently coupled nonlinear Schr\"odinger equations. Varying the relative strength of cross-phase and self-phase effects we show existence and origin of four branches of MI of the two-wave solitary solutions. We give a physical interpretation of our results in terms of the group velocity dispersion (GVD) induced polarization dynamics of spatial solitary waves. In particular, we show that in media with normal GVD spatial symmetry breaking changes to polarization symmetry breaking when the relative strength of the cross-phase modulation exceeds a certain threshold value. The analytical and numerical stability analyses are fully supported by an extensive series of numerical simulations of the full model.Comment: Physical Review E, July, 199

Hyperspherical Harmonics, Separation of Variables and the Bethe Ansatz

The relation between solutions to Helmholtz's equation on the sphere $S^{n-1}$ and the [{\gr sl}(2)]^n Gaudin spin chain is clarified. The joint eigenfuctions of the Laplacian and a complete set of commuting second order operators suggested by the $R$--matrix approach to integrable systems, based on the loop algebra \wt{sl}(2)_R, are found in terms of homogeneous polynomials in the ambient space. The relation of this method of determining a basis of harmonic functions on $S^{n-1}$ to the Bethe ansatz approach to integrable systems is explained.Comment: 14 pgs, Plain Tex, preprint CRM--2174 (May, 1994