The free rigid body dynamics: generalized versus classic

In this paper we analyze the normal forms of a general quadratic Hamiltonian system defined on the dual of the Lie algebra $\mathfrak{o}(K)$ of real $K$ - skew - symmetric matrices, where $K$ is an arbitrary $3\times 3$ real symmetric matrix. A consequence of the main results is that any first-order autonomous three-dimensional differential equation possessing two independent quadratic constants of motion which admits a positive/negative definite linear combination, is affinely equivalent to the classical "relaxed" free rigid body dynamics with linear controls.Comment: 12 page

Kinetic and ion pairing contributions in the dielectric spectra of electrolyte aqueous solutions

Understanding dielectric spectra can reveal important information about the dynamics of solvents and solutes from the dipolar relaxation times down to electronic ones. In the late 1970s, Hubbard and Onsager predicted that adding salt ions to a polar solution would result in a reduced dielectric permittivity that arises from the unexpected tendency of solvent dipoles to align opposite to the applied field. So far, this effect has escaped an experimental verification, mainly because of the concomitant appearance of dielectric saturation from which the Hubbard-Onsager decrement cannot be easily separated. Here we develop a novel non-equilibrium molecular dynamics simulation approach to determine this decrement accurately for the first time. Using a thermodynamic consistent all-atom force field we show that for an aqueous solution containing sodium chloride around 4.8 Mol/l, this effect accounts for 12\% of the total dielectric permittivity. The dielectric decrement can be strikingly different if a less accurate force field for the ions is used. Using the widespread GROMOS parameters, we observe in fact an {\it increment} of the dielectric permittivity rather than a decrement. We can show that this increment is caused by ion pairing, introduced by a too low dispersion force, and clarify the microscopic connection between long-living ion pairs and the appearance of specific features in the dielectric spectrum of the solution

Two-component {CH} system: Inverse Scattering, Peakons and Geometry

An inverse scattering transform method corresponding to a Riemann-Hilbert problem is formulated for CH2, the two-component generalization of the Camassa-Holm (CH) equation. As an illustration of the method, the multi - soliton solutions corresponding to the reflectionless potentials are constructed in terms of the scattering data for CH2.Comment: 22 pages, 3 figures, draft, please send comment

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

Emergent singular solutions of non-local density-magnetization equations in one dimension

We investigate the emergence of singular solutions in a non-local model for a magnetic system. We study a modified Gilbert-type equation for the magnetization vector and find that the evolution depends strongly on the length scales of the non-local effects. We pass to a coupled density-magnetization model and perform a linear stability analysis, noting the effect of the length scales of non-locality on the system's stability properties. We carry out numerical simulations of the coupled system and find that singular solutions emerge from smooth initial data. The singular solutions represent a collection of interacting particles (clumpons). By restricting ourselves to the two-clumpon case, we are reduced to a two-dimensional dynamical system that is readily analyzed, and thus we classify the different clumpon interactions possible.Comment: 19 pages, 13 figures. Submitted to Phys. Rev.

Theory and simulations of rigid polyelectrolytes

We present theoretical and numerical studies on stiff, linear polyelectrolytes within the framework of the cell model. We first review analytical results obtained on a mean-field Poisson-Boltzmann level, and then use molecular dynamics simulations to show, under which circumstances these fail quantitatively and qualitatively. For the hexagonally packed nematic phase of the polyelectrolytes we compute the osmotic coefficient as a function of density. In the presence of multivalent counterions it can become negative, leading to effective attractions. We show that this results from a reduced contribution of the virial part to the pressure. We compute the osmotic coefficient and ionic distribution functions from Poisson-Boltzmann theory with and without a recently proposed correlation correction, and also simulation results for the case of poly(para-phenylene) and compare it to recently obtained experimental data on this stiff polyelectrolyte. We also investigate ion-ion correlations in the strong coupling regime, and compare them to predictions of the recently advocated Wigner crystal theories.Comment: 32 pages, 15 figures, proceedings of the ASTATPHYS-MEX-2001, to be published in Mol. Phy

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

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