Dynamics of kinks in the Ginzburg-Landau equation: Approach to a metastable shape and collapse of embedded pairs of kinks

We consider initial data for the real Ginzburg-Landau equation having two widely separated zeros. We require these initial conditions to be locally close to a stationary solution (the ``kink'' solution) except for a perturbation supported in a small interval between the two kinks. We show that such a perturbation vanishes on a time scale much shorter than the time scale for the motion of the kinks. The consequences of this bound, in the context of earlier studies of the dynamics of kinks in the Ginzburg-Landau equation, [ER], are as follows: we consider initial conditions $v_0$ whose restriction to a bounded interval $I$ have several zeros, not too regularly spaced, and other zeros of $v_0$ are very far from $I$. We show that all these zeros eventually disappear by colliding with each other. This relaxation process is very slow: it takes a time of order exponential of the length of $I$

Persistence of black holes through a cosmological bounce

We discuss whether black holes could persist in a universe which recollapses and then bounces into a new expansion phase. Whether the bounce is of classical or quantum gravitational origin, such cosmological models are of great current interest. In particular, we investigate the mass range in which black holes might survive a bounce and ways of differentiating observationally between black holes formed just after and just before the last bounce. We also discuss the consequences of the universe going through a sequence of dimensional changes as it passes through a bounce.Comment: 8 pages, 1 figur

Semi-Static Hedging Based on a Generalized Reflection Principle on a Multi Dimensional Brownian Motion

On a multi-assets Black-Scholes economy, we introduce a class of barrier options. In this model we apply a generalized reflection principle in a context of the finite reflection group acting on a Euclidean space to give a valuation formula and the semi-static hedge.Comment: Asia-Pacific Financial Markets, online firs

Generating ring currents, solitons, and svortices by stirring a Bose-Einstein condensate in a toroidal trap

We propose a simple stirring experiment to generate quantized ring currents and solitary excitations in Bose-Einstein condensates in a toroidal trap geometry. Simulations of the 3D Gross-Pitaevskii equation show that pure ring current states can be generated efficiently by adiabatic manipulation of the condensate, which can be realized on experimental time scales. This is illustrated by simulated generation of a ring current with winding number two. While solitons can be generated in quasi-1D tori, we show the even more robust generation of hybrid, solitonic vortices (svortices) in a regime of wider confinement. Svortices are vortices confined to essentially one-dimensional dynamics, which obey a similar phase-offset--velocity relationship as solitons. Marking the transition between solitons and vortices, svortices are a distinct class of symmetry-breaking stationary and uniformly rotating excited solutions of the 2D and 3D Gross-Pitaevskii equation in a toroidal trapping potential. Svortices should be observable in dilute-gas experiments.Comment: 8 pages, 4 figures; accepted for publication in J. Phys. B (Letters

Spontaneous soliton formation and modulational instability in Bose-Einstein condensates

The dynamics of an elongated attractive Bose-Einstein condensate in an axisymmetric harmonic trap is studied. It is shown that density fringes caused by self-interference of the condensate order parameter seed modulational instability. The latter has novel features in contradistinction to the usual homogeneous case known from nonlinear fiber optics. Several open questions in the interpretation of the recent creation of the first matter-wave bright soliton train [Strecker {\it et al.} Nature {\bf 417} 150 (2002)] are addressed. It is shown that primary transverse collapse, followed by secondary collapse induced by soliton--soliton interactions, produce bursts of hot atoms at different time scales.Comment: 4 pages, 3 figures. Phys. Rev. Lett. in pres

Signatures of superconducting gap inhomogeneities in optical properties

Scanning tunneling spectroscopy applied to the high-$T_{c}$ cuprates has revealed significant spatial inhomogeneity on the nanoscale. Regions on the order of a coherence length in size show variations of the magnitude of the superconducting gap of order $\pm20$ or more. An important unresolved question is whether or not these variations are also present in the bulk, and how they influence superconducting properties. As many theories and data analyses for high-$T_{c}$ superconductivity assume spatial homogeneity of the gap magnitude, this is a pressing question. We consider the far-infrared optical conductivity and evaluate, within an effective medium approximation, what signatures of spatial variations in gap magnitude are present in various optical quantities. In addition to the case of d-wave superconductivity, relevant to the high-$T_c$ cuprates, we have also considered s-wave gap symmetry in order to provide expected signatures of inhomogeneities for superconductors in general. While signatures of gap inhomogeneities can be strongly manifested in s-wave superconductors, we find that the far-infrared optical conductivity in d-wave is robust against such inhomogeneity.Comment: 8 pages, 7 figure

Shear induced breaking of large internal solitary waves

The stability properties of 24 experimentally generated internal solitary waves (ISWs) of extremely large amplitude, all with minimum Richardson number less than 1/4, are investigated. The study is supplemented by fully nonlinear calculations in a three-layer fluid. The waves move along a linearly stratified pycnocline (depth h2) sandwiched between a thin upper layer (depth h1) and a deep lower layer (depth h3), both homogeneous. In particular, the wave-induced velocity profile through the pycnocline is measured by particle image velocimetry (PIV) and obtained in computation. Breaking ISWs were found to have amplitudes (a1) in the range a1>2.24 √h1h2(1+h2/h1), while stable waves were on or below this limit. Breaking ISWs were investigated for 0.27 0.86 and stable waves for Lx/λ < 0.86. The results show a sort of threshold-like behaviour in terms of Lx/λ. The results demonstrate that the breaking threshold of Lx/λ = 0.86 was sharper than one based on a minimum Richardson number and reveal that the Richardson number was found to become almost antisymmetric across relatively thick pycnoclines, with the minimum occurring towards the top part of the pycnoclinePostprintPeer reviewe

Option Pricing in Multivariate Stochastic Volatility Models of OU Type

We present a multivariate stochastic volatility model with leverage, which is flexible enough to recapture the individual dynamics as well as the interdependencies between several assets while still being highly analytically tractable. First we derive the characteristic function and give conditions that ensure its analyticity and absolute integrability in some open complex strip around zero. Therefore we can use Fourier methods to compute the prices of multi-asset options efficiently. To show the applicability of our results, we propose a concrete specification, the OU-Wishart model, where the dynamics of each individual asset coincide with the popular Gamma-OU BNS model. This model can be well calibrated to market prices, which we illustrate with an example using options on the exchange rates of some major currencies. Finally, we show that covariance swaps can also be priced in closed form.Comment: 28 pages, 5 figures, to appear in SIAM Journal on Financial Mathematic

Effect of Primordial Black Holes on the Cosmic Microwave Background and Cosmological Parameter Estimates

We investigate the effect of non-evaporating primordial black holes (PBHs) on the ionization and thermal history of the universe. X-rays emitted by gas accretion onto PBHs modify the cosmic recombination history, producing measurable effects on the spectrum and anisotropies of the Cosmic Microwave Background (CMB). Using the third-year WMAP data and FIRAS data we improve existing upper limits on the abundance of PBHs with masses >0.1 Msun by several orders of magnitude. Fitting WMAP3 data with cosmological models that do not allow for non-standard recombination histories, as produced by PBHs or other early energy sources, may lead to an underestimate of the best-fit values of the amplitude of linear density fluctuations (sigma_8) and the scalar spectral index (n_s). Cosmological parameter estimates are affected because models with PBHs allow for larger values of the Thomson scattering optical depth, whose correlation with other parameters may not be correctly taken into account when PBHs are ignored. Values of tau_e=0.2, n_s=1 and sigma_8=0.9 are allowed at 95% CF. This result that may relieve recent tension between WMAP3 data and clusters data on the value of sigma_8. PBHs may increase the primordial molecular hydrogen abundance by up to two orders of magnitude, this promoting cooling and star formation. The suppression of galaxy formation due to X-ray heating is negligible for models consistent with the CMB data. Thus, the formation rate of the first galaxies and stars would be enhanced by a population of PBHs.Comment: 17 pages (Apj style), 9 figures, submitted to Ap

Evolution of a Primordial Black Hole Population

We reconsider in this work the effects of an energy absorption term in the evolution of primordial black holes (hereafter PBHs) in the several epochs of the Universe. A critical mass is introduced as a boundary between the accreting and evaporating regimes of the PBHs. We show that the growth of PBHs is negligible in the Radiation-dominated Era due to scarcity of energy density supply from the expanding background, in agreement with a previous analysis by Carr and Hawking, but that nevertheless the absorption term is large enough for black holes above the critical mass to preclude their evaporation until the universe has cooled sufficiently. The effects of PBH motion are also discussed: the Doppler effect may give rise to energy accretion in black-holes with large peculiar motions relative to background. We discuss how cosmological constraints are modified by the introduction of the critical mass since that PBHs above it do not disturb the CMBR. We show that there is a large range of admissible masses for PBHs above the critical mass but well below the cosmological horizon. Finally we outline a minimal kinetic formalism, solved in some limiting cases, to deal with more complicated cases of PBH populationsComment: RevTex file, 8 pp., 3 .ps figures available upon request from [email protected]