DOCSLIB.ORG

Phase-Shift Controlling of Three Solitons in Dispersion-Decreasing fibers

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Nonlinear Dyn https://doi.org/10.1007/s11071-019-05200-5 ORIGINAL PAPER Phase-shift controlling of three solitons in dispersion-decreasing fibers Suzhi Liu · Qin Zhou · Anjan Biswas · Wenjun Liu Received: 3 July 2019 / Accepted: 16 August 2019 © Springer Nature B.V. 2019 Abstract Phase-shift controlling can attenuate the In addition, by adjusting the constraint value, propaga- interactions between solitons and gives practical tion distance of solitons can be further extended. This advantage in optical communication systems. For the may be useful in the optical logic devices and ultra- variable-coefficient nonlinear Schrödinger equation, short pulse lasers. which can be imitated the transmission of solitons in the dispersion-decreasing fiber, analytic three solitons Keywords Soliton · Analytic solution · Dispersion- solutions are derived via the Hirota method. Based on decreasing fiber · Hirota method the obtained solutions, influences of the second-order dispersion parameters and other related parameters in different function types on the soliton transmission 1 Introduction are discussed. Results declare that phase-shift control- ling of solitons in dispersion-decreasing fiber can be Optical solitons, which can be stably transmitted by achieved when the dispersion function is Gaussian one. virtue of the offset about the group velocity disper- sion (GVD) and self-phase modulation (SPM), have S. Liu · W. Liu (B) induced substantial interests of researchers in math- State Key Laboratory of Information Photonics and Optical ematics, physics and optics [1–24]. In particular, the Communications, and School of Science, nonlinearity and integrability in the solitons equation Beijing University of Posts and Telecommunications, P. O. Box 122, Beijing 100876, China are widely concerned [25–28]. Under ideal conditions, e-mail: [email protected] solitons can achieve lossless transmission [29]. How- Q. Zhou ever, when two- or multi-solitons are transmitted simul- School of Electronics and Information Engineering, Wuhan taneously in the optical fiber, these solitons can attract Donghu University, Wuhan 430212, each other [29–31]. Due to the existence of solitary People’s Republic of China interactions, which causes transmission rate of the opti- A. Biswas cal communication system to be seriously degraded, Department of Physics, Chemistry and Mathematics, studying how to minimize the interactions becomes one Alabama A&M University, Normal, AL 35762, USA of the meaningful directions [32,33]. A. Biswas Phase-shift controlling, which contributes to an Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia approach for the question mentioned above, has been investigated widely in theory and experiment [34,35]. A. Biswas Department of Mathematics and Statistics, Tshwane University Ref. [34] has researched large phase shift of solitons in of Technology, Pretoria 0008, South Africa lead glass under strong nonlocal space. Furthermore, 123 S. Liu et al. the phase shifts and trajectories during the overtak- where h(ξ, τ) is assumed as a complex differentiable ing interaction of multi-solitons have been analyzed function while f (ξ, τ) is a real one. After calculation, in Ref. [35]. The propagation of three solitons in the the following bilinear equations of Eq. (1) are obtained: dispersion-decreasing fiber (DDF) can be explained 2 by the nonlinear Schrödinger (NLS) equation as fol- (2iDξ + D(ξ)Dτ )h · f = 0, lows [36,37]: 2 2 g(ξ)dx ∗ D(ξ)Dτ f · f + 2ρ(ξ)e h · h = 0, (3) (ξ) D 2 φξ − i φττ + iρ(ξ)|φ| φ = g(ξ)φ. (1) 2 where * denotes complex conjugate. There is a con- straint relationship among D(ξ), ρ(ξ) and g(ξ): Here, φ is complex function with ξ and τ representing scaled distance and time. D(ξ) is the GVD parame- D(ξ) ter, ρ(ξ) is the Kerr nonlinear parameter, and g(ξ) is ρ(ξ) = P . (4) 2 g(ξ)dξ a parameter related to fiber loss or amplification. With e regard to Eq. (1), the formation and amplification of As Hirota bilinear operators, Dξ and Dτ have the fol- solitons have been studied [38]. Interactions between lowing form: two solitons have been attenuated by phase-shift con- ∂ ∂ m ∂ ∂ n trolling [36]. Besides, researches have done about how Dm Dn(a · b) = − − ξ τ ∂ξ ∂ξ ∂τ ∂τ to amplify, reshape, fission and annihilate solitons [39] and investigated the propagation properties of optical a(ξ, τ)b(ξ ,τ )|ξ =ξ,τ=τ , (5) solitons in the DDF [40]. where both of m and n are non-negative integers, a is the Compared with two solitons, the results of three soli- function about ξ and τ, and b is the function about ξ tons on propagation and interactions are more abun- and τ . To solve bilinear forms (3), h(ξ, τ) and f (ξ, τ) dant. In order to better understand the phase-shift char- can be expanded as: acteristics of three solitons in DDF, results of interac- tions are shown and analyzed under the GVD param- h(ξ, τ) = h (ξ, τ) + 3h (ξ, τ) + 5h (ξ, τ), eters with different functions in this paper. By setting 1 3 5 (ξ, τ) = + 2 (ξ, τ) + 4 (ξ, τ) + 6 (ξ, τ). appropriate dispersion values, phase-shift controlling f 1 f2 f4 f6 of solitons in DDF can be better achieved, which is (6) helpful to improve the transmission quality of the sig- nal in optical communication system. In addition, the Without affecting the generality, we can define the loss in the medium can be effectively compensated value of as 1, and expand the bilinear forms (3)by by changing the constraint value produced by D(ξ), using expression (6). Three solitons solutions of Eq. (1) ρ(ξ) and g(ξ), so that the propagation distance can be are constructed as follows: extended. The structure of the paper is as follows. Section 2 h (ξ, τ) + h (ξ, τ) + h (ξ, τ) (ξ) ξ φ(ξ,τ) = 1 3 5 e g d , is arranged to get analytic three solitons solutions of 1 + f2(ξ, τ) + f4(ξ, τ) + f6(ξ, τ) Eq. (1) through the Hirota method. Section 3 is reserved (7) for analyzing interactions influences under different value of related parameters and finding suitable value to where weaken the interactions among three solitons. Finally, θ1 θ2 θ3 Sect. 4 is used to compose a conclusion for this paper. h1(ξ, τ) = e + e + e , ∗ ∗ θ +θ1 θ +θ1 f2(ξ, τ) = A1(ξ)e 1 + A2(ξ)e 2 ∗ ∗ θ +θ1 θ +θ2 2 Bilinear forms and three solitons solutions + A3(ξ)e 3 + A4(ξ)e 1 ∗ ∗ θ +θ2 θ +θ2 + A5(ξ)e 2 + A6(ξ)e 3 ∗ ∗ θ +θ3 θ +θ3 Using the Hirota method to introduce the dependent + A7(ξ)e 1 + A8(ξ)e 2 ∗ variable transformation: θ +θ3 + A9(ξ)e 3 , θ∗+θ +θ θ∗+θ +θ h (ξ, τ) = B (ξ)e 1 1 2 + B (ξ)e 2 1 2 h(ξ, τ) (ξ) ξ 3 1 2 φ(ξ,τ) = g d , θ∗+θ +θ θ∗+θ +θ e (2) + (ξ) 3 1 2 + (ξ) 1 1 3 f (ξ, τ) B3 e B4 e 123 Phase-shift controlling of three solitons θ∗+θ +θ θ∗+θ +θ 2 2 2 2 2 + (ξ) 2 1 3 + (ξ) 3 1 3 w w P J P B5 e B6 e (ξ) = 41 32 , (ξ) = 21 , θ∗+θ +θ θ∗+θ +θ M4 M5 + (ξ) 1 2 3 + (ξ) 2 2 3 2 w2 w2 w2 2 2 2 B7 e B8 e 4r11 21 22 23 16r11r31 J22 θ∗+θ +θ + (ξ) 3 2 3 , B9 e w2 w2 2 θ∗+θ∗+θ +θ θ∗+θ∗+θ +θ 12 43 P (ξ, τ) = (ξ) 1 2 1 2 + (ξ) 1 3 1 2 (ξ) = , f4 M1 e M2 e M6 2 2 2 2 ∗ ∗ ∗ ∗ 4r w w w θ +θ +θ1+θ2 θ +θ +θ1+θ3 31 31 32 23 + M3(ξ)e 2 3 + M4(ξ)e 1 2 θ∗+θ∗+θ +θ θ∗+θ∗+θ +θ 2 2 2 2 2 2 + (ξ) 1 3 1 3 + (ξ) 2 3 1 3 w w P w w P M5 e M6 e (ξ) = 41 13 , (ξ) = 13 23 , θ∗+θ∗+θ +θ θ∗+θ∗+θ +θ M7 M8 + (ξ) 1 2 2 3 + (ξ) 1 3 2 3 2 w2 w2 w2 2 w2 w2 w2 M7 e M8 e 4r21 21 22 23 4r31 21 33 22 ∗ ∗ θ +θ +θ2+θ3 + M9(ξ)e 2 3 , 2 2 θ∗+θ∗+θ +θ +θ θ∗+θ∗+θ +θ +θ J31 P (ξ, τ) = (ξ) 1 2 1 2 3 + (ξ) 1 3 1 2 3 (ξ) = , h5 N1 e N2 e M9 2 2 2 ∗ ∗ 16r r J θ +θ +θ1+θ2+θ3 21 31 32 + N3(ξ)e 2 3 , ∗ ∗ ∗ θ +θ +θ +θ1+θ2+θ3 2 w2 w2 2 f6(ξ, τ) = n(ξ)e 1 2 3 , J11 12 13 P N1(ξ) = , 16r 2 r 2 w2 w2 w2 w2 with 11 21 31 21 22 23 θ = a (ξ)+ia (ξ)+(r + ir )τ + k + ik , J 2 w2 w2 P2 i i1 i2 i1 i2 i1 i2 N (ξ) = 21 11 13 , 2 2 2 w2 w2 w2 w2 16r11r31 21 32 33 22 ai1(ξ) =− ri1ri2 D(ξ)dξ, 2 w2 w2 2 1 J31 11 12 P (ξ) = ( 2 − 2 ) (ξ) ξ( = , , ), N3(ξ) = , ai2 ri1 ri2 D d i 1 2 3 2 2 w2 w2 w2 w2 2 16r21r31 31 32 33 23 P P P A (ξ) =− , A (ξ) =− , A (ξ) =− , J 2 J 2 J 2 P3 1 2 2 w2 3 w2 n(ξ) =− 11 21 31 , 4r11 31 32 2 2 2 w2 w2 w2 w2 w2 w2 64r11r21r31 31 21 32 33 22 23 P P P A (ξ) =− , A (ξ) =− , A (ξ) =− , w = r + ir − r − ir , 4 w2 5 2 6 w2 11 11 12 21 22 21 4r21 33 w12 = r11 + ir12 − r31 − ir32, P P P A (ξ) =− , A (ξ) =− , A (ξ) =− , w = + − − , 7 w2 8 w2 9 2 13 r21 ir22 r31 ir32 22 23 4r31 w21 = r11 − ir12 + r21 + ir22, w2 P w2 P B (ξ) =− 11 , B (ξ) =− 11 , w = − + + , 1 2 w2 2 2 w2 22 r11 ir12 r31 ir32 4r11 21 4r21 31 w23 = r21 − ir22 + r31 + ir32, w2 P B (ξ) =− 11 , w = + + − , 3 w2 w2 31 r11 ir12 r21 ir22 32 33 w32 = r11 + ir12 + r31 − ir32, w2 P w2 P B (ξ) =− 12 , B (ξ) =− 12 , w = + + − , 4 2 w2 5 w2 w2 33 r21 ir22 r31 ir32 4r11 22 31 23 w41 = r11 − ir12 − r21 + ir22, w2 P B (ξ) =− 12 , w = − − + , 6 2 w2 42 r11 ir12 r31 ir32 4r31 32 w43 = r21 − ir22 − r31 + ir32, w2 P w2 P B (ξ) =− 13 , B (ξ) =− 13 , = ( − )2 + ( − )2, 7 w2 w2 8 2 w2 J11 r11 r21 r12 r22 21 22 4r21 23 2 2 J12 = (r11 + r21) + (r12 − r22) , w2 P B (ξ) =− 13 , = ( − )2 + ( − )2, 9 2 w2 J21 r11 r31 r12 r32 4r31 33 J = (r + r )2 + (r − r )2, 2 2 w2 w2 2 22 11 31 12 32 J11 P 11 42 P M1(ξ) = , M2(ξ) = , J = (r − r )2 + (r − r )2, 16r 2 r 2 J 2 4r 2 w2 w2 w2 31 21 31 22 32 11 21 12 11 21 32 33 2 2 J32 = (r21 + r31) + (r22 − r32) .
Recommended publications
  • Dispersion of Tsunamis: Does It Really Matter? and Physics and Physics Discussions Open Access Open Access S

    Dispersion of Tsunamis: Does It Really Matter? and Physics and Physics Discussions Open Access Open Access S

    EGU Journal Logos (RGB) Open Access Open Access Open Access Advances in Annales Nonlinear Processes Geosciences Geophysicae in Geophysics Open Access Open Access Nat. Hazards Earth Syst. Sci., 13, 1507–1526, 2013 Natural Hazards Natural Hazards www.nat-hazards-earth-syst-sci.net/13/1507/2013/ doi:10.5194/nhess-13-1507-2013 and Earth System and Earth System © Author(s) 2013. CC Attribution 3.0 License. Sciences Sciences Discussions Open Access Open Access Atmospheric Atmospheric Chemistry Chemistry Dispersion of tsunamis: does it really matter? and Physics and Physics Discussions Open Access Open Access S. Glimsdal1,2,3, G. K. Pedersen1,3, C. B. Harbitz1,2,3, and F. Løvholt1,2,3 Atmospheric Atmospheric 1International Centre for Geohazards (ICG), Sognsveien 72, Oslo, Norway Measurement Measurement 2Norwegian Geotechnical Institute, Sognsveien 72, Oslo, Norway 3University of Oslo, Blindern, Oslo, Norway Techniques Techniques Discussions Open Access Correspondence to: S. Glimsdal ([email protected]) Open Access Received: 30 November 2012 – Published in Nat. Hazards Earth Syst. Sci. Discuss.: – Biogeosciences Biogeosciences Revised: 5 April 2013 – Accepted: 24 April 2013 – Published: 18 June 2013 Discussions Open Access Abstract. This article focuses on the effect of dispersion in 1 Introduction Open Access the field of tsunami modeling. Frequency dispersion in the Climate linear long-wave limit is first briefly discussed from a the- Climate Most tsunami modelers rely on the shallow-water equations oretical point of view. A single parameter, denoted as “dis- of the Past for predictions of propagationof and the run-up. Past Some groups, on persion time”, for the integrated effect of frequency dis- Discussions the other hand, insist on applying dispersive wave models, persion is identified.
  • Waves and Weather

    Waves and Weather

    Waves and Weather 1. Where do waves come from? 2. What storms produce good surfing waves? 3. Where do these storms frequently form? 4. Where are the good areas for receiving swells? Where do waves come from? ==> Wind! Any two fluids (with different density) moving at different speeds can produce waves. In our case, air is one fluid and the water is the other. • Start with perfectly glassy conditions (no waves) and no wind. • As wind starts, will first get very small capillary waves (ripples). • Once ripples form, now wind can push against the surface and waves can grow faster. Within Wave Source Region: - all wavelengths and heights mixed together - looks like washing machine ("Victory at Sea") But this is what we want our surfing waves to look like: How do we get from this To this ???? DISPERSION !! In deep water, wave speed (celerity) c= gT/2π Long period waves travel faster. Short period waves travel slower Waves begin to separate as they move away from generation area ===> This is Dispersion How Big Will the Waves Get? Height and Period of waves depends primarily on: - Wind speed - Duration (how long the wind blows over the waves) - Fetch (distance that wind blows over the waves) "SMB" Tables How Big Will the Waves Get? Assume Duration = 24 hours Fetch Length = 500 miles Significant Significant Wind Speed Wave Height Wave Period 10 mph 2 ft 3.5 sec 20 mph 6 ft 5.5 sec 30 mph 12 ft 7.5 sec 40 mph 19 ft 10.0 sec 50 mph 27 ft 11.5 sec 60 mph 35 ft 13.0 sec Wave height will decay as waves move away from source region!!! Map of Mean Wind
  • Basin Scale Tsunami Propagation Modeling Using Boussinesq Models: Parallel Implementation in Spherical Coordinates

    Basin Scale Tsunami Propagation Modeling Using Boussinesq Models: Parallel Implementation in Spherical Coordinates

    WCCE – ECCE – TCCE Joint Conference: EARTHQUAKE & TSUNAMI BASIN SCALE TSUNAMI PROPAGATION MODELING USING BOUSSINESQ MODELS: PARALLEL IMPLEMENTATION IN SPHERICAL COORDINATES J. T. Kirby1, N. Pophet2, F. Shi1, S. T. Grilli3 ABSTRACT We derive weakly nonlinear, weakly dispersive model equations for propagation of surface gravity waves in a shallow, homogeneous ocean of variable depth on the surface of a rotating sphere. A numerical scheme is developed based on the staggered-grid finite difference formulation of Shi et al (2001). The model is implemented using the domain decomposition technique in conjunction with the message passing interface (MPI). The efficiency tests show a nearly linear speedup on a Linux cluster. Relative importance of frequency dispersion and Coriolis force is evaluated in both the scaling analysis and the numerical simulation of an idealized case on a sphere. 1 INTRODUCTION The conventional models in the global-scale tsunami modeling are based on the shallow water equations and neglect frequency dispersion effects in wave propagation. Recent studies on tsunami modeling revealed that such tsunami models may not be satisfactory in predicting tsunamis caused by nonseismic sources (Løvholt et al., 2008). For seismic tsunamis, the frequency dispersion effects in the long distance propagation of tsunami fronts may become significant. The numerical simulations of the 2004 Indian Ocean tsunami by Glimsdal et al. (2006) and Grue et al. (2008) indicated the undular bores may evolve in shallow water, as the phenomenon evidenced in observations (Shuto, 1985). In the simulation for the same tsunami by Grilli et al. (2007), the dispersive effects were quantified by running the dispersive Boussinesq model FUNWAVE (Kirby et al., 1998) and the NSWE solver.
  • Shallow Water Waves and Solitary Waves Article Outline Glossary

    Shallow Water Waves and Solitary Waves Article Outline Glossary

    Shallow Water Waves and Solitary Waves Willy Hereman Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado, USA Article Outline Glossary I. Definition of the Subject II. Introduction{Historical Perspective III. Completely Integrable Shallow Water Wave Equations IV. Shallow Water Wave Equations of Geophysical Fluid Dynamics V. Computation of Solitary Wave Solutions VI. Water Wave Experiments and Observations VII. Future Directions VIII. Bibliography Glossary Deep water A surface wave is said to be in deep water if its wavelength is much shorter than the local water depth. Internal wave A internal wave travels within the interior of a fluid. The maximum velocity and maximum amplitude occur within the fluid or at an internal boundary (interface). Internal waves depend on the density-stratification of the fluid. Shallow water A surface wave is said to be in shallow water if its wavelength is much larger than the local water depth. Shallow water waves Shallow water waves correspond to the flow at the free surface of a body of shallow water under the force of gravity, or to the flow below a horizontal pressure surface in a fluid. Shallow water wave equations Shallow water wave equations are a set of partial differential equations that describe shallow water waves. 1 Solitary wave A solitary wave is a localized gravity wave that maintains its coherence and, hence, its visi- bility through properties of nonlinear hydrodynamics. Solitary waves have finite amplitude and propagate with constant speed and constant shape. Soliton Solitons are solitary waves that have an elastic scattering property: they retain their shape and speed after colliding with each other.
  • Dispersion Coefficients for Coastal Regions

    Dispersion Coefficients for Coastal Regions

    (!1vL- L/0;z7 /- NUREG/CR-3149 PNL-4627 Dispersion Coefficients for Coastal Regions Prepared by B. L. MacRae, R. J. Kaleel, D. L. Shearer/ TRC TRC Environmental Consultants, Inc. Pacific Northwest Laboratory Operated by Battelle Memorial Institute Prepared for U.S. Nuclear Regulatory Commission NOTICE This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, or any of their employees, makes any warranty, expressed or implied, or assumes any legal liability of re ­ sponsibility for any third party's use, or the results of such use, of any information, apparatus, product or process disclosed in this report, or represents that its use by such third party would not infringe privately owned rights. Availability of Reference Materials Cited in NRC Publications Most documents cited in NRC publications will be available from one of the following sources: 1. The NRC Public Document Room, 1717 H Street, N.W. Washington, DC 20555 2 The NRC/GPO Sales Program, U.S. Nuclear Regulatory Commission, Washington, DC 20555 3. The National Technical Information Service, Springfield, VA 22161 Although the listing that follows represents the majority of documents cited in NRC publications, it is not intended to be exhaustive. Referenced documents available for inspection and copying for a fee from the NRC Public Docu­ ment Room include NRC correspondence and ir.ternal NRC memoranda; NRC Office of Inspection and Enforcement bulletins, circulars, information not1ces, inspection and investigation notices; licensee Event Reports; vendor reports and correspondence; Commission papers; and applicant and licensee documents and correspondence.
  • Ethekwini Economic Review 2006 / 2007 Ethekwini Municipality Economic Review 2006/2007

    Ethekwini Economic Review 2006 / 2007 Ethekwini Municipality Economic Review 2006/2007

    eThekwini Economic Review 2006 / 2007 eThekwini Municipality Economic Review 2006/2007 Published by eThekwini Municipality Economic Development Department Research conducted by Researchers: James Dray, Andrew McGill, Graham Muller, Kay Muller, Dayne Skinner Copyright © 2006 eThekwini Municipality Economic Develoment Department PO Box 5856 Durban 4000 KwaZulu-Natal South Africa Telephone: +27 (31) 311 3824 Website: www.durban.gov.za E-mail: [email protected] It should be noted that any opinions expressed are those of the researchers and not of the eThekwini Economic Development Department or the eThekwini Municipality Contents Chapter 1 - eThekwini Economy 3.3 Land Use in Context PG 1 3.4 Demographics 1.0 Introduction 1.1 Global Economic Outlook 3.5 Employment and Unemployment 1.1.1 Growth 1.1.2 Non-oil Commodity Prices 3.6 Education in eThekwini 1.1.3 Oil Prices 3.6.1 Primary and Secondary Schooling 3.6.2 Higher Education 1.2 An Overview of South African Economy 1.2.1 Economic Growth 3.7 An Assessment of Poverty and Inequality in 1.2.2 Macro-economic Stability eThekwini 1.2.3 Infl ation 3.7.1 Measuring Poverty 1.2.4 The Exchange Rate 3.7.2 Income Inequality 1.2.5 The Current Account 1.2.6 The Property Market 3.8 Health 1.2.7 The Equity Market 3.8.1 HIV/AIDS 1.2.8 Savings and Investment 3.8.2 Tuberculosis (TB) 1.2.9 Foreign Direct Investment 3.8.3 Pollution 1.2.10 Forecasts for the South African Economy 3.8.4 Health Care 1.3 The eThekwini Municipality in a National 3.9 Crime Context 1.3.1 Population 3.10 Conclusion 1.3.2 Contribution
  • Directions to K-RITH

    Directions to K-RITH

    Directions to K-RITH Directions from Pietermaritzburg Get on to the N3 towards Durban. Take the Tollgate turnoff. Turn right into Mazisi Kunene Rd (South Ridge Rd). Continue along this road (hilly and twisty), until you come to a T-junction. Turn left into Rick Turner Rd (Francois Rd). At the first set of traffic lights, turn right into ZK Matthews Rd (Nicholson Rd). Continue up the hill. At the traffic circle, turn left into Sphiwe Zuma Rd (Queen Mary Ave). At the second set of traffic lights, turn left into Umbilo Rd. Keep to the right hand lane. Entrance to Nelson R Mandela School of Medicine is 100m on the right. Directions from the South (Amanzimtoti) Get on to the N2 towards Durban. Take the M7 Solomon Mahlanga Drive (Higginson Highway) turnoff. At the traffic lights, turn right. Continue for approximately 3.5 km. Take the R102 Umbilo Rd off ramp. Continue along this road for approximately 2.5 km. Keep to the right. The entrance to the Nelson R Mandela School of Medicine is 100m on the right after the 4th set of traffic lights. Directions from the North (Durban North / City) Take the M4 Ruth First Highway towards Durban. Take the Argyll Rd off ramp. Turn right into Argyll Rd. Continue for approximately 1.6 km. Turn left into Cowey Rd. Continue straight along this road for 6 km, crossing over the N3 highway. At the traffic circle, turn left into Sphiwe Zuma Rd (Queen Mary Ave). At the second set of traffic lights, turn left into Umbilo Rd.
  • Ethekwini Municipality: Draft 2014/15 IDP

    Ethekwini Municipality: Draft 2014/15 IDP

    eThekwini Municipality: Draft 2014/15 IDP eThekwini Municipality Draft Integrated Development Plan 5 Year Plan: 2012/13 to 2016/17 Annual Review 2014/2015 1 eThekwini Municipality: Draft 2014/15 IDP Contents CHAPTER 1: ......................................................................................................................... 4 SITUATIONAL ANALYSIS ..................................................................................................... 4 1.1 POPULATION: ...................................................................................................................................................... 4 1.2 GENDER MAINSTREAMING: .................................................................................................................................... 9 1.3: SOCIAL DEVELOPMENT CHALLENGES ................................................................................................................ 10 1.4 HUMAN CAPITAL DEVELOPMENT ......................................................................................................................... 13 1.5 ECONOMY: ........................................................................................................................................................ 16 1.6 USHAKA MARINE WORLD ................................................................................................................................... 25 1.7 ICC DURBAN ....................................................................................................................................................
  • The Development of an Open Space System for the Queensburgb Municipal Area

    The Development of an Open Space System for the Queensburgb Municipal Area

    THE DEVELOPMENT OF AN OPEN SPACE SYSTEM FOR THE QUEENSBURGB MUNICIPAL AREA. BY KERRY ANN SEPPINGS SUBMITTED IN FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE M.Se IN THE FACULTY OF SCIENCE BIOLOGY DEPARTMENT UNIVERSITY OF NATAL DECEMBER 1994 PREFACE The work described in this thesis was carried out ill the Department of Biology, University of Natal (Durban) from January 1993 to December 1994, under the supervision of Professor J. Cooke. This study represents original work by the author unless specifically stated to the contrary in the text, and has not been submitted in any form to another university. Kerry Ann Seppings December 1994 ACKNOWLEDGEMENTS I would like to extend my sincere gratitude to the following people for their assistance with this study: The Council and staff of Queensburgh Municipality, especially Councillor Gloria Rolando and the Engineers Department, for their support, both fmancial and otherwise and their assistance in the field. Dr R. Westfall for his time and computer expertise and the use of his computer program package. My mom and dad for their financial and moral support throughout this project and for their assistance with the many public events hosted to promote the Queensburgh Open Space System. My sister for all the proof reading and colouring in of maps. Oliver for all his help in the field (despite the many confrontations with extensive stands of Chromolaena and Lantana) and the many, many hours spent promoting environmental awareness and helping with the public events. Prof. Cooke for his advice and guidance in the compilation of this thesis. As always - Dr Debra Roberts for initially supervising this project and continued advice and encouragement.
  • Solitons Et Dispersion

    Solitons Et Dispersion

    Solitons et dispersion Raphaël Côte Table des matières Table des matières i Introduction iii Preamble: Orbital stability of solitons vii Solitons . vii Orbital stability of solitons . ix 1 Multi-solitons 1 1 Construction of multi-solitons . 1 2 Families of multi-solitons . 8 2 Soliton resolution for wave type equations 11 1 Profile decomposition . 12 2 Linear energy outside the light cone . 14 3 The nonlinear wave equation in dimension 3 . 17 4 Equivariant wave maps to the sphere S2 ....................... 19 5 Back to the H˙ 1 × L2 critical wave equation: dimension 4 . 23 3 Blowup for the 1D semi linear wave equation 27 1 Blow up and blow up curve . 27 2 Description of the blow up . 28 3 Construction of characteristic points . 33 4 The Zakharov-Kuznetsov flow around solitons 37 1 Liouville theorems . 39 2 Asymptotic stability . 41 3 Multi-solitons . 42 5 Formation of Néel walls 45 1 A two-dimensional model for thin-film micromagnetism . 45 2 Compactness and optimality of Néel walls . 48 3 Formation of static Néel walls . 49 Index 53 Bibliography 55 References presented for the habilitation . 55 General references . 56 i Introduction objet de ce mémoire est de présenter quelques aspects de la dynamique des solutions L’ d’équations aux dérivées partielles dispersives. Le fil conducteur liant les différents travaux exposés ici est l’étude des solitons. Ce sont des ondes progressives ou stationnaires, des objets non linéaires qui préservent leur forme au cours du temps. Elles possèdent des propriétés de rigidité remarquables, et jouent un rôle prééminent dans la description de solutions génériques en temps long.
  • Dispersion of Wave Action Abstract Introduction

    Dispersion of Wave Action Abstract Introduction

    Dispersion of Wave Action James M. Buick1 and Tariq S. Durrani2 Abstract The dispersion of particles beneath a random sea, produced by the associated random variations in the Stokes drift is considered. In particular, a sea state described by a Pierson-Moskowitz spectrum is considered and the diffusion rate on and below the surface is examined. For such a random sea the diffusion rate at the surface can be related to the wind speed. This expression is extended to consider dispersion rates below the surface and to relates these to the wind speed. The theory is then applied to study the dispersion of particles denser than water as they fall through a random sea. Introduction An accurate prediction of the dispersion of pollutants in the sea is essential in the planning of contingency exercises and also in risk analysis. Computer models have been developed to simulate the spread of pollutants in the sea but these generally ignore the effects of wave action in dispersing the material which, on a scale of up to a few kilometers, can have a large role to play (Schott et al. 1978, Herterich and Hasselmann 1982). Recent oil spill disasters such as the sinking of the Braer off the Shetland Islands have highlighted the fact that the dispersion of pollutants is highly dependent on the prevailing wave climate. Discharges 'Research Fellow, Signal Processing Division, Department of Electronic and Electrical Engineering, Uni- versity of Strathclyde, Glasgow Gl 1XQ, UK. Present address: Department of Physics and Astronomy, The University of Edinburgh, J.C.M.B., Mayfield Road, Edinburgh EH9 3JZ, U.K, [email protected] 2Professor,Signal Processing Division, Department of Electronic and Electrical Engineering, University of Strathclyde, Glasgow Gl 1XQ, UK 3553 3554 COASTAL ENGINEERING 1998 from chemical and sewage works can also cause pollution.
  • Swell and Wave Forecasting

    Swell and Wave Forecasting

    Lecture 24 Part II Swell and Wave Forecasting 29 Swell and Wave Forecasting • Motivation • Terminology • Wave Formation • Wave Decay • Wave Refraction • Shoaling • Rouge Waves 30 Motivation • In Hawaii, surf is the number one weather-related killer. More lives are lost to surf-related accidents every year in Hawaii than another weather event. • Between 1993 to 1997, 238 ocean drownings occurred and 473 people were hospitalized for ocean-related spine injuries, with 77 directly caused by breaking waves. 31 Going for an Unintended Swim? Lulls: Between sets, lulls in the waves can draw inexperienced people to their deaths. 32 Motivation Surf is the number one weather-related killer in Hawaii. 33 Motivation - Marine Safety Surf's up! Heavy surf on the Columbia River bar tests a Coast Guard vessel approaching the mouth of the Columbia River. 34 Sharks Cove Oahu 35 Giant Waves Peggotty Beach, Massachusetts February 9, 1978 36 Categories of Waves at Sea Wave Type: Restoring Force: Capillary waves Surface Tension Wavelets Surface Tension & Gravity Chop Gravity Swell Gravity Tides Gravity and Earth’s rotation 37 Ocean Waves Terminology Wavelength - L - the horizontal distance from crest to crest. Wave height - the vertical distance from crest to trough. Wave period - the time between one crest and the next crest. Wave frequency - the number of crests passing by a certain point in a certain amount of time. Wave speed - the rate of movement of the wave form. C = L/T 38 Wave Spectra Wave spectra as a function of wave period 39 Open Ocean – Deep Water Waves • Orbits largest at sea sfc.