Vortex line representation for flows of ideal and viscous fluids

It is shown that the Euler hydrodynamics for vortical flows of an ideal fluid coincides with the equations of motion of a charged {\it compressible} fluid moving due to a self-consistent electromagnetic field. Transition to the Lagrangian description in a new hydrodynamics is equivalent for the original Euler equations to the mixed Lagrangian-Eulerian description - the vortex line representation (VLR). Due to compressibility of a "new" fluid the collapse of vortex lines can happen as the result of breaking (or overturning) of vortex lines. It is found that the Navier-Stokes equation in the vortex line representation can be reduced to the equation of the diffusive type for the Cauchy invariant with the diffusion tensor given by the metric of the VLR

LOFAR observations of fine spectral structure dynamics in type IIIb radio bursts

Solar radio emission features a large number of fine structures demonstrating great variability in frequency and time. We present spatially resolved spectral radio observations of type IIIb bursts in the $30-80$ MHz range made by the Low Frequency Array (LOFAR). The bursts show well-defined fine frequency structuring called "stria" bursts. The spatial characteristics of the stria sources are determined by the propagation effects of radio waves; their movement and expansion speeds are in the range of 0.1-0.6c. Analysis of the dynamic spectra reveals that both the spectral bandwidth and the frequency drift rate of the striae increase with an increase of their central frequency; the striae bandwidths are in the range of ~20-100 kHz and the striae drift rates vary from zero to ~0.3 MHz s^-1. The observed spectral characteristics of the stria bursts are consistent with the model involving modulation of the type III burst emission mechanism by small-amplitude fluctuations of the plasma density along the electron beam path. We estimate that the relative amplitude of the density fluctuations is of dn/n~10^-3, their characteristic length scale is less than 1000 km, and the characteristic propagation speed is in the range of 400-800 km/s. These parameters indicate that the observed fine spectral structures could be produced by propagating magnetohydrodynamic waves

New boundary conditions for integrable lattices

New boundary conditions for integrable nonlinear lattices of the XXX type, such as the Heisenberg chain and the Toda lattice are presented. These integrable extensions are formulated in terms of a generic XXX Heisenberg magnet interacting with two additional spins at each end of the chain. The construction uses the most general rank 1 ansatz for the 2x2 L-operator satisfying the reflection equation algebra with rational r-matrix. The associated quadratic algebra is shown to be the one of dynamical symmetry for the A1 and BC2 Calogero-Moser problems. Other physical realizations of our quadratic algebra are also considered.Comment: 22 pages, latex, no figure

Interaction of a vortex ring with the free surface of ideal fluid

The interaction of a small vortex ring with the free surface of a perfect fluid is considered. In the frame of the point ring approximation the asymptotic expression for the Fourier-components of radiated surface waves is obtained in the case when the vortex ring comes from infinity and has both horizontal and vertical components of the velocity. The non-conservative corrections to the equations of motion of the ring, due to Cherenkov radiation, are derived.Comment: LaTeX, 15 pages, 1 eps figur

Formation of singularities on the surface of a liquid metal in a strong electric field

The nonlinear dynamics of the free surface of an ideal conducting liquid in a strong external electric field is studied. It is establish that the equations of motion for such a liquid can be solved in the approximation in which the surface deviates from a plane by small angles. This makes it possible to show that on an initially smooth surface for almost any initial conditions points with an infinite curvature corresponding to branch points of the root type can form in a finite time.Comment: 14 page

Zipf's Law in Gene Expression

Using data from gene expression databases on various organisms and tissues, including yeast, nematodes, human normal and cancer tissues, and embryonic stem cells, we found that the abundances of expressed genes exhibit a power-law distribution with an exponent close to -1, i.e., they obey Zipf's law. Furthermore, by simulations of a simple model with an intra-cellular reaction network, we found that Zipf's law of chemical abundance is a universal feature of cells where such a network optimizes the efficiency and faithfulness of self-reproduction. These findings provide novel insights into the nature of the organization of reaction dynamics in living cells.Comment: revtex, 11 pages, 3 figures, submitted to Phys. Rev. Let

Field induced evolution of regular and random 2D domain structures and shape of isolated domains in LiNbO₃ and LiTaO₃

The shapes of isolated domains produced by application of the uniform external electric field in different experimental conditions were investigated experimentally in single crystalline lithium niobate LiNbO3 and lithium tantalate LiTaO3. The study of the domain kinetics by computer simulation and experimentally by polarization reversal of the model structure using two-dimensional regular electrode pattern confirms applicability of the kinetic approach to explanation of the experimentally observed evolution of the domain shape and geometry of the domain structure. It has been shown that the fast domain walls strictly oriented along X directions appear after domain merging

Commuting difference operators with elliptic coefficients from Baxter's vacuum vestors

For quantum integrable models with elliptic R-matrix, we construct the Baxter Q-operator in infinite-dimensional representations of the algebra of observables.Comment: 31 pages, LaTeX, references adde