Variational Principles for Lagrangian Averaged Fluid Dynamics

The Lagrangian average (LA) of the ideal fluid equations preserves their transport structure. This transport structure is responsible for the Kelvin circulation theorem of the LA flow and, hence, for its convection of potential vorticity and its conservation of helicity. Lagrangian averaging also preserves the Euler-Poincar\'e (EP) variational framework that implies the LA fluid equations. This is expressed in the Lagrangian-averaged Euler-Poincar\'e (LAEP) theorem proven here and illustrated for the Lagrangian average Euler (LAE) equations.Comment: 23 pages, 3 figure

Complete integrability versus symmetry

The purpose of this article is to show that on an open and dense set, complete integrability implies the existence of symmetry

GG-Strands

A $G$-strand is a map $g(t,{s}):\,\mathbb{R}\times\mathbb{R}\to G$ for a Lie group $G$ that follows from Hamilton's principle for a certain class of $G$-invariant Lagrangians. The SO(3)-strand is the $G$-strand version of the rigid body equation and it may be regarded physically as a continuous spin chain. Here, $SO(3)_K$-strand dynamics for ellipsoidal rotations is derived as an Euler-Poincar\'e system for a certain class of variations and recast as a Lie-Poisson system for coadjoint flow with the same Hamiltonian structure as for a perfect complex fluid. For a special Hamiltonian, the $SO(3)_K$-strand is mapped into a completely integrable generalization of the classical chiral model for the SO(3)-strand. Analogous results are obtained for the $Sp(2)$-strand. The $Sp(2)$-strand is the $G$-strand version of the $Sp(2)$ Bloch-Iserles ordinary differential equation, whose solutions exhibit dynamical sorting. Numerical solutions show nonlinear interactions of coherent wave-like solutions in both cases. ${\rm Diff}(\mathbb{R})$-strand equations on the diffeomorphism group $G={\rm Diff}(\mathbb{R})$ are also introduced and shown to admit solutions with singular support (e.g., peakons).Comment: 35 pages, 5 figures, 3rd version. To appear in J Nonlin Sc

Coupling of nitrogen-vacancy centers in diamond to a GaP waveguide

The optical coupling of guided modes in a GaP waveguide to nitrogen-vacancy (NV) centers in diamond is demonstrated. The electric field penetration into diamond and the loss of the guided mode are measured. The results indicate that the GaP-diamond system could be useful for realizing coupled microcavity-NV devices for quantum information processing in diamond.Comment: 4 pages 4 figure

Strong Attraction between Charged Spheres due to Metastable Ionized States

We report a mechanism which can lead to long range attractions between like-charged spherical macroions, stemming from the existence of metastable ionized states. We show that the ground state of a single highly charged colloid plus a few excess counterions is overcharged. For the case of two highly charged macroions in their neutralizing divalent counterion solution we demonstrate that, in the regime of strong Coulomb coupling, the counterion clouds are very likely to be unevenly distributed, leading to one overcharged and one undercharged macroion. This long-living metastable configuration in turn leads to a long range Coulomb attraction.Comment: REVTEX-published versio

Calculation of critical exponents by self-similar factor approximants

The method of self-similar factor approximants is applied to calculating the critical exponents of the O(N)-symmetric phi^4 theory and of the Ising glass. It is demonstrated that this method, being much simpler than other known techniques of series summation in calculating the critical exponents, at the same time, yields the results that are in very good agreement with those of other rather complicated numerical methods. The principal advantage of the method of self-similar factor approximants is the combination of its extraordinary simplicity and high accuracy.Comment: 17 pages, 2 table

Fluctuation effects of gauge fields in the slave-boson t-J model

We present a quantitative study of the charge-spin separation(CSS) phenomenon in a U(1) gauge theory of the t-J model of high-Tc superconductures. We calculate the critical temperature of confinement-deconfinement phase transition below which the CSS takes place.Comment: Latex, 9 pages, 3 figure

Mean-Field HP Model, Designability and Alpha-Helices in Protein Structures

Analysis of the geometric properties of a mean-field HP model on a square lattice for protein structure shows that structures with large number of switch backs between surface and core sites are chosen favorably by peptides as unique ground states. Global comparison of model (binary) peptide sequences with concatenated (binary) protein sequences listed in the Protein Data Bank and the Dali Domain Dictionary indicates that the highest correlation occurs between model peptides choosing the favored structures and those portions of protein sequences containing alpha-helices.Comment: 4 pages, 2 figure

An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion

We study a class of 1+1 quadratically nonlinear water wave equations that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation, yet still preserves integrability via the inverse scattering transform (IST) method. This IST-integrable class of equations contains both the KdV equation and the CH equation as limiting cases. It arises as the compatibility condition for a second order isospectral eigenvalue problem and a first order equation for the evolution of its eigenfunctions. This integrable equation is shown to be a shallow water wave equation derived by asymptotic expansion at one order higher approximation than KdV. We compare its traveling wave solutions to KdV solitons.Comment: 4 pages, no figure

Integrable discretizations of some cases of the rigid body dynamics

A heavy top with a fixed point and a rigid body in an ideal fluid are important examples of Hamiltonian systems on a dual to the semidirect product Lie algebra $e(n)=so(n)\ltimes\mathbb R^n$. We give a Lagrangian derivation of the corresponding equations of motion, and introduce discrete time analogs of two integrable cases of these systems: the Lagrange top and the Clebsch case, respectively. The construction of discretizations is based on the discrete time Lagrangian mechanics on Lie groups, accompanied by the discrete time Lagrangian reduction. The resulting explicit maps on $e^*(n)$ are Poisson with respect to the Lie--Poisson bracket, and are also completely integrable. Lax representations of these maps are also found.Comment: arXiv version is already officia