Some New Addition Formulae for Weierstrass Elliptic Functions

We present new addition formulae for the Weierstrass functions associated with a general elliptic curve. We prove the structure of the formulae in n-variables and give the explicit addition formulae for the 2- and 3-variable cases. These new results were inspired by new addition formulae found in the case of an equianharmonic curve, which we can now observe as a specialisation of the results here. The new formulae, and the techniques used to find them, also follow the recent work for the generalisation of Weierstrass' functions to curves of higher genus.Comment: 20 page

Optimal escape from circular orbits around black holes

Using the theory of optimal rocket trajectories in general relativity, recently developed in arXiv:1105.5235, we show that the "obvious" manoeuvre of using a tangential instantaneous acceleration to escape a stable circular orbit in the Schwarzschild spacetime satisfies the optimality conditions if and only if the magnitude of the acceleration is smaller than a certain bound.Comment: 7 page

Acceleration and localization of matter in a ring trap

A toroidal trap combined with external time-dependent electric field can be used for implementing different dynamical regimes of matter waves. In particular, we show that dynamical and stochastic acceleration, localization and implementation of the Kapitza pendulum can be originated by means of proper choice of the external force

An analytical study of resonant transport of Bose-Einstein condensates

We study the stationary nonlinear Schr\"odinger equation, or Gross-Pitaevskii equation, for a one--dimensional finite square well potential. By neglecting the mean--field interaction outside the potential well it is possible to discuss the transport properties of the system analytically in terms of ingoing and outgoing waves. Resonances and bound states are obtained analytically. The transmitted flux shows a bistable behaviour. Novel crossing scenarios of eigenstates similar to beak--to--beak structures are observed for a repulsive mean-field interaction. It is proven that resonances transform to bound states due to an attractive nonlinearity and vice versa for a repulsive nonlinearity, and the critical nonlinearity for the transformation is calculated analytically. The bound state wavefunctions of the system satisfy an oscillation theorem as in the case of linear quantum mechanics. Furthermore, the implications of the eigenstates on the dymamics of the system are discussed.Comment: RevTeX4, 16 pages, 19 figure

Optimal time travel in the Godel universe

Using the theory of optimal rocket trajectories in general relativity, recently developed in arXiv:1105.5235, we present a candidate for the minimum total integrated acceleration closed timelike curve in the Godel universe, and give evidence for its minimality. The total integrated acceleration of this curve is lower than Malament's conjectured value (Malament, 1984), as was already implicit in the work of Manchak (Manchak, 2011); however, Malament's conjecture does seem to hold for periodic closed timelike curves.Comment: 16 pages, 2 figures; v2: lower bound in the velocity and reference adde

On the generalized continuity equation

A generalized continuity equation extending the ordinary continuity equation has been found using quanternions. It is shown to be compatible with Dirac, Schrodinger, Klein-Gordon and diffusion equations. This generalized equation is Lorentz invariant. The transport properties of electrons are found to be governed by Schrodinger-like equation and not by the diffusion equation.Comment: 9 Latex pages, no figure

Integration of the primer vector in a central force field

This paper examines the primer vector which governs optimal solutions for orbital transfer when the central force field has a more general form than the usual inverse-square-force law. Along a null-thrust are that connects two successive impulses, the two sets of state and adjoint equations are decoupled. This allows the reduction of the problem to the integration of a linear first-order differential equation, and hence the solution of the optimal coasting are in the most general central force field can be obtained by simple quadratures. Immediate applications of the results can be seen in solving problems of escape in the equatorial plane of an oblate planet, satellite swing by, or station keeping around Lagrangian points in the three-body problem.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45248/1/10957_2004_Article_BF00932804.pd

The probability of an encounter of two Brownian particles before escape

We study the probability of two Brownian particles to meet before one of them exits a finite interval. We obtain an explicit expression for the probability as a function of the initial distance of the two particles using the Weierstrass elliptic function. We also find the law of the meeting location. Brownian simulations show the accuracy of our analysis. Finally, we discuss some applications to the probability that a double strand DNA break repairs in confined environments.Comment: To appear J. Phys

Effected of Feshbach resonance on dynamics of matter waves in optical lattices

Mean-filed dynamics of a Bose-Einstein condensate (BEC) loaded in an optical lattice (OL), confined by a parabolic potentials, and subjected to change of a scattering length by means of the Feshbach resonance (FR), is considered. The system is described by the Gross-Pitaevskii (GP) equation with varying nonlinearity, which in a number of cases can be reduced a one-dimensional perturbed nonlinear Schr\"{o}dinger (NLS) equation. A particular form of the last one depends on relations among BEC parameters. We describe periodic solutions of the NLS equation and their adiabatic dynamics due to varying nonlinearity; carry out numerical study of the dynamics of the NLS equation with periodic and parabolic trap potentials. We pay special attention to processes of generation of trains of bright and dark matter solitons from initially periodic waves.Comment: 16 pages, 11 figures (revised version). to be published in Phys. Rev. A (2005

The Picard-Fuchs equations for complete hyperelliptic integrals of even order curves, and the actions of the generalized Neumann system

We consider a family of genus 2 hyperelliptic curves of even order and obtain explicitly the systems of 5 linear ordinary differential equations for periods of the corresponding Abelian integrals of first, second, and third kind, as functions of some parameters of the curves. The systems can be regarded as extensions of the well-studied Picard-Fuchs equations for periods of complete integrals of first and second kind on odd hyperelliptic curves. The periods we consider are linear combinations of the action variables of several integrable systems, in particular the generalized Neumann system with polynomial separable potentials. Thus the solutions of the extended Picard-Fuchs equations can be used to study various properties of the actions. (C) 2014 AIP Publishing LLC.Peer ReviewedPostprint (published version