Bifurcations of periodic orbits with spatio-temporal symmetries

Motivated by recent analytical and numerical work on two- and three-dimensional convection with imposed spatial periodicity, we analyse three examples of bifurcations from a continuous group orbit of spatio-temporally symmetric periodic solutions of partial differential equations. Our approach is based on centre manifold reduction for maps, and is in the spirit of earlier work by Iooss (1986) on bifurcations of group orbits of spatially symmetric equilibria. Two examples, two-dimensional pulsating waves (PW) and three-dimensional alternating pulsating waves (APW), have discrete spatio-temporal symmetries characterized by the cyclic groups Z_n, n=2 (PW) and n=4 (APW). These symmetries force the Poincare' return map M to be the nth iterate of a map G: M=G^n. The group orbits of PW and APW are generated by translations in the horizontal directions and correspond to a circle and a two-torus, respectively. An instability of pulsating waves can lead to solutions that drift along the group orbit, while bifurcations with Floquet multiplier +1 of alternating pulsating waves do not lead to drifting solutions. The third example we consider, alternating rolls, has the spatio-temporal symmetry of alternating pulsating waves as well as being invariant under reflections in two vertical planes. This leads to the possibility of a doubling of the marginal Floquet multiplier and of bifurcation to two distinct types of drifting solutions. We conclude by proposing a systematic way of analysing steady-state bifurcations of periodic orbits with discrete spatio-temporal symmetries, based on applying the equivariant branching lemma to the irreducible representations of the spatio-temporal symmetry group of the periodic orbit, and on the normal form results of Lamb (1996). This general approach is relevant to other pattern formation problems, and contributes to our understanding of the transition from ordered to disordered behaviour in pattern-forming systems

Hydrodynamics of confined colloidal fluids in two dimensions

We apply a hybrid Molecular Dynamics and mesoscopic simulation technique to study the dynamics of two dimensional colloidal discs in confined geometries. We calculate the velocity autocorrelation functions, and observe the predicted $t^{-1}$ long time hydrodynamic tail that characterizes unconfined fluids, as well as more complex oscillating behavior and negative tails for strongly confined geometries. Because the $t^{-1}$ tail of the velocity autocorrelation function is cut off for longer times in finite systems, the related diffusion coefficient does not diverge, but instead depends logarithmically on the overall size of the system.Comment: RevTex 13 pages, 9 figure

Symmetry of Dirac Equation and Corresponding Phenomenology

It has been suggested that the high symmetries in the Schr\"odinger equation with the Coulomb or harmonic oscillator potentials may remain in the corresponding relativistic Dirac equation. If the principle is correct, in the Dirac equation the potential should have a form as ${(1+\beta)\over 2}V(r)$ where $V(r)$ is ${-e^2\over r}$ for hydrogen atom and $\kappa r^2$ for harmonic oscillator. However, in the case of hydrogen atom, by this combination the spin-orbit coupling term would not exist and it is inconsistent with the observational spectra of hydrogen atom, so that the symmetry of SO(4) must reduce into SU(2). The governing mechanisms QED and QCD which induce potential are vector-like theories, so at the leading order only vector potential exists. However, the higher order effects may cause a scalar fraction. In this work, we show that for QED, the symmetry restoration is very small and some discussions on the symmetry breaking are made. At the end, we briefly discuss the QCD case and indicate that the situation for QCD is much more complicated and interesting.Comment: 15pages, 3 figures, accepted by International Journal of Modern Physics

A photon transport problem with a time-dependent point source

We consider a time-dependent problem of photon transport in an interstellar cloud with a point photon source modeled by a Dirac δ functional. The existence of a unique distributional solution to this problem is established by using the theory of continuous semigroups of operators on locally convex spaces coupled with a constructive approach for producing spaces of generalized functions

Diffuse Gamma-ray Emission from the Galactic Center - A Multiple Energy Injection Model

We suggest that the energy source of the observed diffuse gamma-ray emission from the direction of the Galactic center is the Galactic black hole Sgr A*, which becomes active when a star is captured at a rate of $\sim 10^{-5}$ yr^{-1}. Subsequently the star is tidally disrupted and its matter is accreted into the black hole. During the active phase relativistic protons with a characteristic energy $\sim 6\times 10^{52}$ erg per capture are ejected. Over 90% of these relativistic protons disappear due to proton-proton collisions on a timescale $\tau_{pp} \sim 10^4$ years in the small central bulge region with radius $\sim 50$ pc within Sgr A*, where the density is $\ge 10^3$ cm^{-3}. The gamma-ray intensity, which results from the decay of neutral pions produced by proton-proton collisions, decreases according to $e^{-t/\tau_{pp}}$, where t is the time after last stellar capture. Less than 5% of relativistic protons escaped from the central bulge region can survive and maintain their energy for >10^7 years due to much lower gas density outside, where the gas density can drop to $\sim 1$ cm$^{-3}$. They can diffuse to a $\sim 500$ pc region before disappearing due to proton-proton collisions. The observed diffuse GeV gamma-rays resulting from the decay of neutral pions produced via collision between these escaped protons and the gas in this region is expected to be insensitive to time in the multi-injection model with the characteristic injection rate of 10^{-5} yr^{-1}. Our model calculated GeV and 511 keV gamma-ray intensities are consistent with the observed results of EGRET and INTEGRAL, however, our calculated inflight annihilation rate cannot produce sufficient intensity to explain the COMPTEL data.Comment: 8 pages, 3 figures, accepted by A&

Symmetries and reversing symmetries of toral automorphisms

Toral automorphisms, represented by unimodular integer matrices, are investigated with respect to their symmetries and reversing symmetries. We characterize the symmetry groups of GL(n,Z) matrices with simple spectrum through their connection with unit groups in orders of algebraic number fields. For the question of reversibility, we derive necessary conditions in terms of the characteristic polynomial and the polynomial invariants. We also briefly discuss extensions to (reversing) symmetries within affine transformations, to PGL(n,Z) matrices, and to the more general setting of integer matrices beyond the unimodular ones.Comment: 34 page

Imaginary Phases in Two-Level Model with Spontaneous Decay

We study a two-level model coupled to the electromagnetic vacuum and to an external classic electric field with fixed frequency. The amplitude of the external electric field is supposed to vary very slow in time. Garrison and Wright [{\it Phys. Lett.} {\bf A128} (1988) 177] used the non-hermitian Hamiltonian approach to study the adiabatic limit of this model and obtained that the probability of this two-level system to be in its upper level has an imaginary geometric phase. Using the master equation for describing the time evolution of the two-level system we obtain that the imaginary phase due to dissipative effects is time dependent, in opposition to Garrison and Wright result. The present results show that the non-hermitian hamiltonian method should not be used to discuss the nature of the imaginary phases in open systems.Comment: 11 pages, new version, to appear in J. Phys.

Wigner Distribution Function Approach to Dissipative Problems in Quantum Mechanics with emphasis on Decoherence and Measurement Theory

We first review the usefulness of the Wigner distribution functions (WDF), associated with Lindblad and pre-master equations, for analyzing a host of problems in Quantum Optics where dissipation plays a major role, an arena where weak coupling and long-time approximations are valid. However, we also show their limitations for the discussion of decoherence, which is generally a short-time phenomenon with decay rates typically much smaller than typical dissipative decay rates. We discuss two approaches to the problem both of which use a quantum Langevin equation (QLE) as a starting-point: (a) use of a reduced WDF but in the context of an exact master equation (b) use of a WDF for the complete system corresponding to entanglement at all times

Stochastic Process Associated with Traveling Wave Solutions of the Sine-Gordon Equation

Stochastic processes associated with traveling wave solutions of the sine-Gordon equation are presented. The structure of the forward Kolmogorov equation as a conservation law is essential in the construction and so is the traveling wave structure. The derived stochastic processes are analyzed numerically. An interpretation of the behaviors of the stochastic processes is given in terms of the equation of motion.Comment: 12 pages, 9 figures; corrected typo

Particle trajectories in linearized irrotational shallow water flows

We investigate the particle trajectories in an irrotational shallow water flow over a flat bed as periodic waves propagate on the water's free surface. Within the linear water wave theory, we show that there are no closed orbits for the water particles beneath the irrotational shallow water waves. Depending on the strength of underlying uniform current, we obtain that some particle trajectories are undulating path to the right or to the left, some are looping curves with a drift to the right and others are parabolic curves or curves which have only one loop