Exact Moving and Stationary Solutions of a Generalized Discrete Nonlinear Schrodinger Equation

We obtain exact moving and stationary, spatially periodic and localized solutions of a generalized discrete nonlinear Schr\"odinger equation. More specifically, we find two different moving periodic wave solutions and a localized moving pulse solution. We also address the problem of finding exact stationary solutions and, for a particular case of the model when stationary solutions can be expressed through the Jacobi elliptic functions, we present a two-point map from which all possible stationary solutions can be found. Numerically we demonstrate the generic stability of the stationary pulse solutions and also the robustness of moving pulses in long-term dynamics.Comment: 22 pages, 7 figures, to appear in J. Phys.

Folding of Cu, Zn superoxide dismutase and Familial Amyotrophic Lateral Sclerosis

Cu,Zn superoxide dismutase (SOD1) has been implicated in the familial form of the neurodegenerative disease Amyotrophic Lateral Sclerosis (ALS). It has been suggested that mutant mediated SOD1 misfolding/aggregation is an integral part of the pathology of ALS. We study the folding thermodynamics and kinetics of SOD1 using a hybrid molecular dynamics approach. We reproduce the experimentally observed SOD1 folding thermodynamics and find that the residues which contribute the most to SOD1 thermal stability are also crucial for apparent two-state folding kinetics. Surprisingly, we find that these residues are located on the surface of the protein and not in the hydrophobic core. Mutations in some of the identified residues are found in patients with the disease. We argue that the identified residues may play an important role in aggregation. To further characterize the folding of SOD1, we study the role of cysteine residues in folding and find that non-native disulfide bond formation may significantly alter SOD1 folding dynamics and aggregation propensity.Comment: 16 pages, 5 figure

New Shape Invariant Potentials in Supersymmetric Quantum Mechanics

Quantum mechanical potentials satisfying the property of shape invariance are well known to be algebraically solvable. Using a scaling ansatz for the change of parameters, we obtain a large class of new shape invariant potentials which are reflectionless and possess an infinite number of bound states. They can be viewed as q-deformations of the single soliton solution corresponding to the Rosen-Morse potential. Explicit expressions for energy eigenvalues, eigenfunctions and transmission coefficients are given. Included in our potentials as a special case is the self-similar potential recently discussed by Shabat and Spiridonov.Comment: 8pages, Te

Chemical Enrichment at High Redshifts

We have tried to understand the recent observations related to metallicity in Ly $\alpha$ forest clouds in the framework of the two component model suggested by Chiba & Nath (1997). We find that even if the mini-halos were chemically enriched by an earlier generation of stars, to have [C/H] $\simeq$ -2.5, the number of C IV lines with column density $>10^{12} cm^{-2}$, contributed by the mini-halos, at the redshift of 3, would be only about 10% of the total number of lines, for a chemical enrichment rate of $(1+z)^{-3}$ in the galaxies. Recently reported absence of heavy element lines associated with most of the Ly $\alpha$ lines with H I column density between $10^{13.5} cm^{-2}$ and $10^{14} cm^{-2}$ by Lu et al (1998), if correct, gives an upper limit on [C/H]=-3.7, not only in the mini-halos, but also in the outer parts of galactic halos. This is consistent with the results of numerical simulations, according to which, the chemical elements associated with the Ly $\alpha$ clouds are formed in situ in clouds, rather than in an earlier generation of stars. However, the mean value of $7 \times 10^{-3}$ for the column density ratio of C IV and H I, determined by Cowie and Songaila (1998) for low Lyman alpha optical depths, implies an abundance of [C/H] =-2.5 in mini-halos as well as in most of the region in galactic halos, presumably enriched by an earlier generation of stars. The redshift and column density distribution of C IV has been shown to be in reasonable agreement with the observations.Comment: 23 pages, 6 figures, To appear in Astrophysical Journa

Faces of weight polytopes and a generalization of a theorem of Vinberg

The paper is motivated by the study of graded representations of Takiff algebras, cominuscule parabolics, and their generalizations. We study certain special subsets of the set of weights (and of their convex hull) of the generalized Verma modules (or GVM's) of a semisimple Lie algebra \lie g. In particular, we extend a result of Vinberg and classify the faces of the convex hull of the weights of a GVM. When the GVM is finite-dimensional, we ask a natural question that arises out of Vinberg's result: when are two faces the same? We also extend the notion of interiors and faces to an arbitrary subfield \F of the real numbers, and introduce the idea of a weak \F-face of any subset of Euclidean space. We classify the weak \F-faces of all lattice polytopes, as well as of the set of lattice points in them. We show that a weak \F-face of the weights of a finite-dimensional \lie g-module is precisely the set of weights lying on a face of the convex hull.Comment: Statement changed in Section 4. Typos fixed and some proofs updated. Submitted to "Algebra and Representation Theory." 18 page

Algebra of pion sources

Assuming that the space integrals of source terms (sources) in the Klein-Gordon equation for the pion fields together with isospin generators form an SU(2)⊗SU(2) algebra which, in the soft-pion limit, is a good symmetry of the strong interactions, we calculate S-wave scattering lengths for the collision of pions with hadron targets as well as the pion-nucleon coupling constant. The results are in excellent agreement with experiment

Superposition in nonlinear wave and evolution equations

Real and bounded elliptic solutions suitable for applying the Khare-Sukhatme superposition procedure are presented and used to generate superposition solutions of the generalized modified Kadomtsev-Petviashvili equation (gmKPE) and the nonlinear cubic-quintic Schroedinger equation (NLCQSE).Comment: submitted to International Journal of Theoretical Physics, 23 pages, 2 figures, style change