Annette L. Worthy

Solitary wave evolution for mKdV equations

The evolution of an initial condition into soliton(s) is the classic problem for the Korteweg-de Vries (KdV) equation. While this evolution is theoretically given by the inverse scattering solution of the KdV equation, in practice only the final steady state can be easily obtained from inverse scattering. However, an approximate method based on the conservation laws for the KdV equation has been found to give very accurately the evolution of an initial condition into soliton( s) . This approximate method also gives a criterion for the number of solitons formed. In the present work, this method is extended to describe the evolution of an initial condition into solitary wave(s) for mKdV equations, these equations having the same dispersive term as the KdV equation, but a nonlinear term of the form U”U x, where n 2 1 is a positive integer. It is found that for n < 4, the behaviour of the mKdV equation is similar to the KdV equation in that solitary wave(s) evolve from an arbitrary initial condition. However, for n 2 4, it is found that an initial condition of sufficiently small amplitude decays into dispersive radiation with no solitary wave being formed. For an initial amplitude exceeding a threshold, it is found that the amplitude blows-up. The solutions of the approximate equations are compared with full numerical solutions of the mKdV equation and good agreement is found.

Pulse evolution for a two-dimensional Sine-Gordon equation

The evolution lump and ring solutions of a Sine-Gordon equation in two-space dimensions is considered. Approximate equations governing this evolution are derived using a pulse or ring with variable parameters in an averaged Lagrangian for the Sine-Gordon equation. It was found by Neu [Physica D 43 (1990) 421] that angular variations of the pulse shape may stabilise it. However, no study of the radiation produced by the pulse was available. In the present work, the coupling of the pulse to the shed radiation is considered. It is shown both asymptotically and numerically that the angular dependence produces spiral waves which shed angular momentum, leading to the ultimate collapse of the pulse. Good quantitative agreement between the asymptotic and numerical solutions is found. In addition, it is shown how the results of the present work can be applied to the Baby Skyrme model. In this regard, it is shown how the non-zero degree of solutions of the Baby Skyrme model prevents the collapse of a non-zero degree pulse shedding zero degree radiation. It is also indicated how the present results could be applied to the study of vortex models. The analysis presented in this work shows how complicated behaviour due to radiation of angular momentum can be captured in simple terms by approximate equations for the relevant degrees of freedom.

Optimization of soliton amplitude in dispersion- decreasing nonlinear optical fibers

The compression of a cw into a periodic train of noninteracting solitons by a dispersion-decreasing fiber is investigated with a variational method. To model the evolution from the cw to the soliton train, an elliptic-function-based expression is used as the trial function in the averaged Lagrangian. Both a continuous dispersion variation and a step dispersion variation in the fiber are considered. By use of an optimization method based on the approximate variational equations, the optimal dispersion profile required for achieving maximum pulse compression in a fixed length of fiber is determined. The solutions of the approximate equations are compared with full numerical solutions of the governing nonlinear Schrodinger equation, and good agreement is found.

Analysis of a Chemostat Model with Variable Yield Coefficient and Substrate Inhibition: Contois Growth Kinetics

We analyze the steady-state operation of a generalized reactor model that encompasses a continuous flow bioreactor and an idealized continuous flow membrane reactor as limiting cases. The biochemical reaction kinetics is governed by a Contois growth model subject to noncompetitive substrate inhibition with a variable substrate yield coefficient. The steady-state performance of the reactor is predicted and stability of the steady-state solutions as a function of dimensionless residence time reported. Our results identified two cases of practical interest. The first feature corresponds to the case where solutions to both no-washout and washout conditions are bistable. The second feature identifies the parameter region in which periodic solutions can occur when the yield coefficient is not constant. Both these features are often undesirable in practical applications and must be avoided. Scaling of the model equations reveals that both the second-bifurcation parameters are functions of the influent concentration. Our results predict how the reactor behavior varies as a function of influent concentration and identify the range of influent concentrations where the reactor displays neither periodic nor bistable behavior.

Analysis of a chemostat model with variable yield coefficient: Contois kinetics

Food processing wastewaters and slurries typically contain high concentrations of biodegradable organic matter. Before the wastewater can be discharged, the pollutant concentration must be reduced. One way to achieve this is by using a biological species (biomass) that consumes the organic matter (substrate). We investigate an unstructured kinetic model for a bioreactor with a variable yield coefficient, taking into account the death rate of the microorganisms. The growth rate is given by a Contois expression, which is often used to model the growth of biomass in wastewaters containing biodegradable organic materials. The analysis shows that the system has natural oscillations for some ranges of the parameters. We also investigate the effects of the death rate parameter on the region of periodic behaviour. References Ajbar, A., Al Ahmad, M. and Ali, E. (2010) On the dynamics of biodegradation of wastewater in aerated continuous bioreactors. Mathematical and Computer Modelling, 54 (9--10):1930--1942. doi:10.1016/j.mcm.2011.04.035 Nelson, M.I. and Sidhu, H.S., (2007). Reducing the emission of pollutants in food processing wastewaters. Chemical Engineering and Processing, 46 (5):429--436. doi:10.1016/j.cep.2006.04.012 Contois, D.E. (1959). Kinetics of bacterial growth: relationship between population density and specific growth rate of continuous cultures. Journal of general microbiology, 21:40--50. doi:10.1099/00221287-21-1-40 Alqahtani, R. T., M. I. Nelson, and A. L. Worthy. (2012). A fundamental analysis of continuous flow bioreactor models with recycle around each reactor governed by Contois kinetics. III. Two and three reactor cascades. Chemical Engineering Journal 183 (0):422-432. doi:10.1016/j.cej.2011.12.061 Abdurahman, N. H., Y. M. Rosli, and N. H. Azhari. (2011). Development of a membrane anaerobic system (MAS) for palm oil mill effluent (POME) treatment. Desalination 266 (1-3):208-212. doi:10.1016/j.desal.2010.08.028 Hu, W.C., Thayanithy, K., Forster, C.F. (2002). A kinetic study of the anaerobic digestion of ice-cream wastewater. process biochemistry, 37:965--971. doi:10.1016/S0032-9592(01)00310-7 Bard, E., (2002). Simulating, Analyzing, and Animating Dynamical Systems: Society for Industrial and Applied Mathematics. doi:10.1137/1.9780898718195

A BIOLOGICAL TREATMENT OF INDUSTRIAL WASTEWATERS: CONTOIS KINETICS

This paper analyses the steady-state operation of a generalized bioreactor model that encompasses a continuous-flow bioreactor and an idealized continuous-flow membrane bioreactor as limiting cases. A biodegradation of organic materials is modelled using Contois growth kinetics. The bioreactor performance is analysed by finding the steady-state solutions of the model and determining their stability as a function of the dimensionless residence time. We show that an effective recycle parameter improves the performance of the bioreactor at moderate values of the dimensionless residence time. However, at sufficiently large values of the dimensionless residence time, the performance of the bioreactor is independent of the recycle ratio. doi:10.1017/S144618111500005X

Comparison of performance due to guided hyperlearning, unguided hyperlearning, and conventional learning in mathematics: an empirical study

In this paper, the use of guided hyperlearning, unguided hyperlearning, and conventional learning methods in mathematics are compared. The design of the research involved a quasi-experiment with a modified single-factor multiple treatment design comparing the three learning methods, guided hyperlearning, unguided hyperlearning, and conventional learning. The participants were from three first-year university classes, numbering 115 students in total. Each group received guided, unguided, or conventional learning methods in one of the three different topics, namely number systems, functions, and graphing. The students’ academic performance differed according to the type of learning. Evaluation of the three methods revealed that only guided hyperlearning and conventional learning were appropriate methods for the psychomotor aspects of drawing in the graphing topic. There was no significant difference between the methods when learning the cognitive aspects involved in the number systems topic and the functions topic.

The Demon Drink

We provide a qualitative analysis of a system of nonlinear differential equations that model the spread of alcoholism through a population. Alcoholism is viewed as an infectious disease and the model treats it within a SIR framework. The model exhibits two generic types of steady-state diagram. The first of these is qualitatively the same as the steady-state diagram in the standard sir model. The second exhibits a backwards transcritical bifurcation. As a consequence of this, there is a region of bistability in which a population of problem drinkers can be sustained, even when the reproduction number is less than one. We obtain a succinct formula for this scenario when the transition between these two cases occurs. doi:10.1017/S1446181117000347

Raman induced destabilisation of asymmetric coupled solitary waves in nonlinear twin-core fibres

The evolution of pulses in both single core fibres and in nonlinear fibre couplers when the effect of Raman scattering is included is studied using an approximate technique based on conservation and moment equations. It is found that in order to obtain good agreement with numerical solutions, the effect of the dispersive radiation shed by the pulses as they evolve must be included. The dominant contribution of Raman scattering to this shed radiation is shelves of low frequency radiation lying behind the pulses. Indeed it is found that these shelves of radiation are the main effect driving the evolution of the pulses. The most dramatic effect of the mass, momentum and energy shed by the pulses to these shelves is the de-stabilisation of the asymmetric coupled solitary wave state in a nonlinear fibre coupler, which corresponds to the loss of the switching characteristics of the coupler.

Wind forcing of edge waves on a truncated exponential shelf

The effects of wind forcing on high frequency edge waves over a truncated exponential shelf is examined. Two wind stress models are used and a comparison of results is made. Also, the results are compared to those obtained for the semi-infinite continental shelf profile.