Ricci magnetic geodesic motion of vortices and lumps

Ricci magnetic geodesic (RMG) motion in a k\"ahler manifold is the analogue of geodesic motion in the presence of a magnetic field proportional to the ricci form. It has been conjectured to model low-energy dynamics of vortex solitons in the presence of a Chern-Simons term, the k\"ahler manifold in question being the $n$-vortex moduli space. This paper presents a detailed study of RMG motion in soliton moduli spaces, focusing on the cases of hyperbolic vortices and spherical $\mathbb{C}P^1$ lumps. It is shown that RMG flow localizes on fixed point sets of groups of holomorphic isometries, but that the flow on such submanifolds does not, in general, coincide with their intrinsic RMG flow. For planar vortices, it is shown that RMG flow differs from an earlier reduced dynamics proposed by Kim and Lee, and that the latter flow is ill-defined on the vortex coincidence set. An explicit formula for the metric on the whole moduli space of hyperbolic two-vortices is computed (extending an old result of Strachan's), and RMG motion of centred two-vortices is studied in detail. Turning to lumps, the moduli space of static $n$-lumps is $Rat_n$, the space of degree $n$ rational maps, which is known to be k\"ahler and geodesically incomplete. It is proved that $Rat_1$ is, somewhat surprisingly, RMG complete (meaning that that the initial value problem for RMG motion has a global solution for all initial data). It is also proved that the submanifold of rotationally equivariant $n$-lumps, $Rat_n^{eq}$, a topologically cylindrical surface of revolution, is intrinsically RMG incomplete for $n=2$ and all $n\geq 5$, but that the extrinsic RMG flow on $Rat_2^{eq}$ (defined by the inclusion $Rat_2^{eq}\hookrightarrow Rat_2$) is complete

The geodesic approximation for lump dynamics and coercivity of the Hessian for harmonic maps

The most fruitful approach to studying low energy soliton dynamics in field theories of Bogomol'nyi type is the geodesic approximation of Manton. In the case of vortices and monopoles, Stuart has obtained rigorous estimates of the errors in this approximation, and hence proved that it is valid in the low speed regime. His method employs energy estimates which rely on a key coercivity property of the Hessian of the energy functional of the theory under consideration. In this paper we prove an analogous coercivity property for the Hessian of the energy functional of a general sigma model with compact K\"ahler domain and target. We go on to prove a continuity property for our result, and show that, for the CP^1 model on S^2, the Hessian fails to be globally coercive in the degree 1 sector. We present numerical evidence which suggests that the Hessian is globally coercive in a certain equivariance class of the degree n sector for n>1. We also prove that, within the geodesic approximation, a single CP^1 lump moving on S^2 does not generically travel on a great circle.Comment: 29 pages, 1 figure; typos corrected, references added, expanded discussion of the main function spac

Quantum lump dynamics on the two-sphere

It is well known that the low-energy classical dynamics of solitons of Bogomol'nyi type is well approximated by geodesic motion in M_n, the moduli space of static n-solitons. There is an obvious quantization of this dynamics wherein the wavefunction evolves according to the Hamiltonian H_0 equal to (half) the Laplacian on M_n. Born-Oppenheimer reduction of analogous mechanical systems suggests, however, that this simple Hamiltonian should receive corrections including k, the scalar curvature of M_n, and C, the n-soliton Casimir energy, which are usually difficult to compute, and whose effect on the energy spectrum is unknown. This paper analyzes the spectra of H_0 and two corrections to it suggested by work of Moss and Shiiki, namely H_1=H_0+k/4 and H_2=H_1+C, in the simple but nontrivial case of a single CP^1 lump moving on the two-sphere. Here M_1=TSO(3), a noncompact kaehler 6-manifold invariant under an SO(3)xSO(3) action, whose geometry is well understood. The symmetry gives rise to two conserved angular momenta, spin and isospin. A hidden isometry of M_1 is found which implies that all three energy spectra are symmetric under spin-isospin interchange. The Casimir energy is found exactly on the zero section of TSO(3), and approximated numerically on the rest of M_1. The lowest 19 eigenvalues of H_i are found for i=0,1,2, and their spin-isospin and parity compared. The curvature corrections in H_1 lead to a qualitatively unchanged low-level spectrum while the Casimir energy in H_2 leads to significant changes. The scaling behaviour of the spectra under changes in the radii of the domain and target spheres is analyzed, and it is found that the disparity between the spectra of H_1 and H_2 is reduced when the target sphere is made smaller.Comment: 35 pages, 3 figure

Instability of breathers in the topological discrete sine-Gordon system

It is demonstrated that the breather solutions recently discovered in the weakly coupled topological discrete sine-Gordon system are spectrally unstable. This is in contrast with more conventional spatially discrete systems, whose breathers are always spectrally stable at sufficiently weak coupling.Comment: 7 page

Breathers in the weakly coupled topological discrete sine-Gordon system

Existence of breather (spatially localized, time periodic, oscillatory) solutions of the topological discrete sine-Gordon (TDSG) system, in the regime of weak coupling, is proved. The novelty of this result is that, unlike the systems previously considered in studies of discrete breathers, the TDSG system does not decouple into independent oscillator units in the weak coupling limit. The results of a systematic numerical study of these breathers are presented, including breather initial profiles and a portrait of their domain of existence in the frequency-coupling parameter space. It is found that the breathers are uniformly qualitatively different from those found in conventional spatially discrete systems.Comment: 19 pages, 4 figures. Section 4 (numerical analysis) completely rewritte

Translationally invariant nonlinear Schrodinger lattices

Persistence of stationary and traveling single-humped localized solutions in the spatial discretizations of the nonlinear Schrodinger (NLS) equation is addressed. The discrete NLS equation with the most general cubic polynomial function is considered. Constraints on the nonlinear function are found from the condition that the second-order difference equation for stationary solutions can be reduced to the first-order difference map. The discrete NLS equation with such an exceptional nonlinear function is shown to have a conserved momentum but admits no standard Hamiltonian structure. It is proved that the reduction to the first-order difference map gives a sufficient condition for existence of translationally invariant single-humped stationary solutions and a necessary condition for existence of single-humped traveling solutions. Other constraints on the nonlinear function are found from the condition that the differential advance-delay equation for traveling solutions admits a reduction to an integrable normal form given by a third-order differential equation. This reduction also gives a necessary condition for existence of single-humped traveling solutions. The nonlinear function which admits both reductions defines a two-parameter family of discrete NLS equations which generalizes the integrable Ablowitz--Ladik lattice.Comment: 24 pages, 4 figure

Incidence of lactic acidosis toxicity among patients on stavudine or zidovudine containing antiretroviral therapy at Lighthouse clinics

Although stavudine and zidovudine remain frequently used in low-income countries in Africa, they are associated with long-term toxicities. Lactic acidosis is one of the most serious toxicities in antiretroviral treatment (ART) and occurs predominantly in regimens containing stavudine (D4T) or zidovudine (AZT). We conducted this study to determine the incidence and risk factors for lactic acidosis among HIV-positive patients that have been on ART for at least 6 months. This study will bridge the gap that exists due to scarcity of data on the extent of toxicities due to long-term use of D4T and AZT. We conducted a retrospective cohort study using routine clinic data from the Lighthouse and Martin Preuss Centre electronic data systems. We used the clinic data collected between 1st January 2004 and 31st December 2011. We included into the analysis all patients that have been on D4T- or AZT-containing ARV drugs for at least 6 months. We analysed the data using Poisson regression of the number of cases of lactic acidosis (LA) on gender, age at ART initiation, baseline BMI, and lipodystrophy in order to determine the incidence and risk factors for lactic acidosis. All statistical analyses were done at 5% significance level. We identified 14,854 patients that have ever been on D4T- or AZT-containing ARV drugs for longer than 5 months. Of these, 43% were male and median age was 34 years. The total number of cases of confirmed LA was 342 with observed mortality rate 40% more than the patients without confirmed LA. There were 23.02 cases of LA for every 1000 patient-years on D4T- or AZT-containing ART regimens. The strongest risk factor identified for developing LA was having a baseline BMI >25 with incidence rate ratio (IRR) 3.11 (95% CI: 2.49, 3.88). The IRR for patients with a diagnosis of lipodystrophy was 1.77 (95% CI: 1.35, 2.32). Patients aged <30 years at ART initiation had 31% reduced risk of developing LA as compared to patients aged>39 years at ART initiation. We were unable to detect any increased risk associated with gender. Clinicians should always have significantly higher index of suspicion of LA in patients with established lipodystrophy, aged more than 30 years at ART initiation and patients with higher baseline BMIs. The number of cases of fatal lactic acidosis that did not present to the clinic is unknown but is likely to be significant

Surface water flood forecasting for urban communities

Key findings and recommendations: • This research has addressed the challenge of surface water flood forecasting by producing the UK’s first operational surface water flood risk forecast with a 24-hour lead time. This was successfully used in Glasgow at the Commonwealth Games in 2014. • The methodology of the Glasgow Pilot has been developed to use nationally available datasets and a transferrable approach which will help urban areas in Scotland improve their resilience to and preparedness for future flooding. • It also delivered a novel method for forecasting the impacts of flooding in real-time and increased knowledge on communicating uncertainties in flood risk. • A real-time forecasting system for surface water flooding from intense rainfall needs to use models that represent surface runoff production, surface water inundation and movement, and how water travels via surface and sub-surface pathways, including urban sewerage and drainage networks. Ensemble rainfall prediction models are key to quantifying uncertainty in forecasting the rainfall that causes surface water flooding. • Detailed surface water flood inundation models exist and are widely used in design and research activities, but none were found to be ready for real-time use. The Grid-to-Grid (G2G) distributed hydrological model was chosen for used in the Glasgow Pilot as it can provide ensemble forecasts of surface water flooding, and takes account of the intensity and pattern of rainfall, land cover and slope, and antecedent conditions. • The research developed a novel methodology for impact assessment that links surface runoff to the severity of flooding impacts on people, property and transport. Use is made of a library of information based on SEPA’s Regional Pluvial (rainfall-related) Flood Hazard maps. • For the Glasgow Pilot, G2G was operated over a 10km by 10km area encompassing Glasgow’s East End and the main areas of activity for the 2014 Commonwealth Games. The research team developed an operational application, called FEWS Glasgow, to support running the model in real-time and reporting on the likely impacts of surface water flooding. A new Daily Glasgow Daily Surface Water Flood Forecast was designed and produced based on operational requirements and emergency responder feedback