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Selected Works

Tim Marchant

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Full-Text Articles in Physics

Reorientational Versus Kerr Dark And Gray Solitary Waves Using Modulation Theory, Prof. Tim Marchant Dec 2010

Reorientational Versus Kerr Dark And Gray Solitary Waves Using Modulation Theory, Prof. Tim Marchant

Tim Marchant

We develop a modulation theory model based on a Lagrangian formulation to investigate the evolution of dark and gray optical spatial solitary waves for both the defocusing nonlinear Schrodinger (NLS) equation and the nematicon equations describing nonlinear beams, nematicons, in self-defocusing nematic liquid crystals. Since it has an exact soliton solution, the defocusing NLS equation is used as a test bed for the modulation theory applied to the nematicon equations, which have no exact solitary wave solution. We find that the evolution of dark and gray NLS solitons, as well as nematicons, is entirely driven by the emission of diffractive ...


The Analytical Evolution Of Nls Solitons Due To The Numerical Discretization Error, Prof. Tim Marchant Dec 2010

The Analytical Evolution Of Nls Solitons Due To The Numerical Discretization Error, Prof. Tim Marchant

Tim Marchant

Soliton perturbation theory is used to obtain analytical solutions describing solitary wave tails or shelves, due to numerical discretization error, for soliton solutions of the nonlinear Schrodinger equation. Two important implicit numerical schemes for the nonlinear Schrodinger equation, with second-order temporal and spatial discretization errors, are considered. These are the Crank-Nicolson scheme and a scheme, due to Taha [1], based on the inverse scattering transform. The first-order correction for the solitary wave tail, or shelf, is in integral form and an explicit expression is found for large time. The shelf decays slowly, at a rate of t(-1/2), which ...


Evolution Of Solitary Waves For A Perturbed Nonlinear Schrodinger Equation, Tim Marchant Dec 2009

Evolution Of Solitary Waves For A Perturbed Nonlinear Schrodinger Equation, Tim Marchant

Tim Marchant

Soliton perturbation theory is used to determine the evolution of a solitary wave described by a perturbed nonlinear Schrödinger equation. Perturbation terms, which model wide classes of physically relevant perturbations, are considered. An analytical solution is found for the first-order correction of the evolving solitary wave. This solution for the solitary wave tail is in integral form and an explicit expression is found, for large time. Singularity theory, usually used for combustion problems, is applied to the large time expression for the solitary wave tail. Analytical results are obtained, such as the parameter regions in which qualitatively different types of ...


Soliton Perturbation Theory For A Higher-Order Hirota Equation, Tim Marchant Dec 2008

Soliton Perturbation Theory For A Higher-Order Hirota Equation, Tim Marchant

Tim Marchant

Solitary wave evolution for a higher order Hirota equation is examined. For the higher order Hirota equation resonance between the solitary waves and linear radiation causes radiation loss. Soliton perturbation theory is used to determine the details of the evolving wave and its tail. An analytical expression for the solitary wave tail is derived and compared to numerical solutions. An excellent comparison between numerical and theoretical solutions is obtained for both right- and left-moving waves. Also, a two-parameter family of higher order asymptotic embedded solitons is identified.


Modulation Analysis Of Boundary Induced Motion Of Optical Solitary Waves In A Nematic Liquid Crystal, Tim Marchant Dec 2008

Modulation Analysis Of Boundary Induced Motion Of Optical Solitary Waves In A Nematic Liquid Crystal, Tim Marchant

Tim Marchant

We consider the motion of a solitary wave, a nematicon, in a finite cell filled with a nematic liquid crystal. A modulation theory is developed to describe the boundary-induced bouncing of a nematicon in a one-dimensional cell and it is found to give predictions in very good agreement with numerical solutions. The boundary-induced motion is then considered numerically for a two-dimensional cell and a simple extension of the modulation theory from one to two space dimensions is then made, with good agreement being found with numerical solutions for the nematicon trajectory. The role of nematicon shape and relative position to ...


A Perturbation Drbem Model For Weakly Nonlinear Wave Run-Ups Around Islands, Tim Marchant Dec 2008

A Perturbation Drbem Model For Weakly Nonlinear Wave Run-Ups Around Islands, Tim Marchant

Tim Marchant

In this paper, the dual reciprocity boundary element method (DRBEM) based on the perturbation method is presented for calculating run-ups of weakly nonlinear long waves scattered by islands. Under the assumption that the incident waves are harmonic, the time-dependent nonlinear Boussinesq equations are transformed into three time-independent linear equations by using the perturbation method, where, besides nonlinearity ε, the dispersion μ2 is also included in the perturbed expansion. The first-order solution η0 is found by using the linear long-wave equations. Then η0 is used in two second-order governing equations related to the dispersion and nonlinearity, respectively. Since no any omission ...


Evolution Of Higher-Order Gray Hirota Solitary Waves, Tim Marchant Dec 2007

Evolution Of Higher-Order Gray Hirota Solitary Waves, Tim Marchant

Tim Marchant

The defocusing Hirota equation has dark and gray soliton solutions which are stable on a background of periodic waves of constant amplitude. In this paper, gray solitary wave evolution for a higher-order defocusing Hirota equation is examined. A direct analysis is used to identify families of higher-order gray Hirota solitary waves, which are embedded for certain parameter values. Soliton perturbation theory is used to detmine the detailed behavior of an evolving higher-order gray Hirota solitary wave. An integral expression for the first-order correction to the wave is found and analytical expressions for the steady-state and transient components of the solitary ...


Mathematical Modelling Of Nematicons And Their Interactions, Prof. Tim Marchant Dec 2007

Mathematical Modelling Of Nematicons And Their Interactions, Prof. Tim Marchant

Tim Marchant

The mathematical modelling of guided wave (nematicon) propagation in liquid crystals is considered. Model equations are derived based on suitable trial functions in an averaged Lagrangian. These equations are used to model nematicon interactions.


Undular Bores And The Initial-Boundary Value Problem For The Modified Korteweg-De Vries Equation, Tim Marchant Dec 2007

Undular Bores And The Initial-Boundary Value Problem For The Modified Korteweg-De Vries Equation, Tim Marchant

Tim Marchant

Two types of analytical undular bore solutions, of the initial value problem for the modified Korteweg-de Vries (mKdV), are found. The first, an undular bore composed of cnoidal waves, is qualitatively similar to the bore found for the KdV equation, with solitons occurring at the leading edge and small amplitude linear waves occurring at the trailing edge. The second, a newly identified type of undular bore, consists of finite amplitude sinusiodal waves, which have a rational form. At the leading edge is the mKdV algebraic soliton, while, again, small amplitude linear waves occur at the trailing edge. The initial-boundary value ...


Collisionless Shock Resolution In Nematic Liquid Crystals, Tim Marchant Dec 2007

Collisionless Shock Resolution In Nematic Liquid Crystals, Tim Marchant

Tim Marchant

The diffractive resolution on a collisionless shock formed along the spatial profile of a beam in a nematic liquid crystal is considered, this material being an example of a self-focusing, nonlocal medium. It is found that the shock is resolved through the formation of an undular bore structure which persists for experimentally relevant propagation distances due to nonlocality delaying the onset of modulational instability. Both 1+1 and 2+1 dimensional bores with circular symmetry are considered (termed line and circular bores, respectively). A semianalytical solution is developed for the line undular bore, approximating it as a train of uniform ...


Semi-Analytical Solutions For A Gray-Scott Reaction-Diffusion Cell With An Applied Electric Field, Tim Marchant Dec 2007

Semi-Analytical Solutions For A Gray-Scott Reaction-Diffusion Cell With An Applied Electric Field, Tim Marchant

Tim Marchant

An ionic version of the Gray–Scott chemical reaction scheme is considered in a reaction–diffusion cell, with an applied electric field, which causes migration of the reactant and autocatalyst in a preferred direction. The Galerkin method is used to reduce the governing partial differential equations to an approximate model consisting of ordinary differential equations. This is accomplished by approximating the spatial structure of the reactant and autocatalyst concentrations. Bifurcation analysis of the semi-analytical model is performed by using singularity theory to analyse the static multiplicity and a stability analysis to determine the dynamic multiplicity. The application of the electric ...


Nonlocal Validity Of An Asymptotic One-Dimensional Nematicon Solution, Tim Marchant Dec 2007

Nonlocal Validity Of An Asymptotic One-Dimensional Nematicon Solution, Tim Marchant

Tim Marchant

The propagation of coherent, polarized light in a nematic liquid crystal, governed by the nematicon equations, is considered. It is found that in the special case of 1 + 1 dimensions and the highly nonlocal limit, the nematicon equations have an asymptotic bulk solitary wave solution, termed a nematicon, which is given in terms of Bessel functions. This asymptotic solution gives both the ground state and the symmetric and antisymmetric excited states, which have multiple peaks. Numerical simulations of nematicon evolution, for parameters corresponding to experimental scenarios, are presented. It is found, for experimentally reasonable parameter choices, that the validity of ...


Dipole Soliton Formation In A Nematic Liquid Crystal In The Non-Local Limit, Tim Marchant Dec 2007

Dipole Soliton Formation In A Nematic Liquid Crystal In The Non-Local Limit, Tim Marchant

Tim Marchant

The interaction of two symmetric solitary waves, termed nematicons, in a liquid crystal is considered in the limit of nonlocal response of the liquid crystal. This nonlocal limit is the applicable limit for most experimentally available liquid crystals. In this nonlocal limit, two separate cases for the initial separation of the nematicons are considered, these being large and small separation. Both spinning and nonspinning nematicons are considered. It is found that in the case of large initial separation, the nematicons can form a spinning or nonspinning bound state with a finite steady separation, this being called a nematicon dipole, when ...


Self-Heating In Compost Piles Due To Biological Effects, Tim Marchant Dec 2006

Self-Heating In Compost Piles Due To Biological Effects, Tim Marchant

Tim Marchant

The increase in temperature in compost piles/landfill sites due to micro-organisms undergoing exothermic reactions is modelled. A simplified model is considered in which only biological self-heating is present. The heat release rate due to biological activity is modelled by a function which is a monotonic increasing function of temperature over the range 0⩽T⩽a, whilst for T⩾a it is a monotone decreasing function of temperature. This functional dependence represents the fact that micro-organisms die or become dormant at high temperatures. The bifurcation behaviour is investigated for 1-d slab and 2-d rectangular slab geometries. In both cases there ...


Solitary Wave Interaction For A Higher-Order Nonlinear Schrodinger Equation, Tim Marchant Dec 2006

Solitary Wave Interaction For A Higher-Order Nonlinear Schrodinger Equation, Tim Marchant

Tim Marchant

Solitary wave interaction for a higher-order version of the nonlinear Schrödinger (NLS) equation is examined. An asymptotic transformation is used to transform a higher-order NLS equation to a higher-order member of the NLS integrable hierarchy, if an algebraic relationship between the higher-order coefficients is satisfied. The transformation is used to derive the higher-order one- and two-soliton solutions; in general, the N-soliton solution can be derived. It is shown that the higher-order collision is asymptotically elastic and analytical expressions are found for the higher-order phase and coordinate shifts. Numerical simulations of the interaction of two higher-order solitary waves are also performed ...


Asymptotic Solitons On A Non-Zero Mean Level., Tim Marchant Dec 2006

Asymptotic Solitons On A Non-Zero Mean Level., Tim Marchant

Tim Marchant

The collision of solitary waves for a higher-order modified Korteweg-de Vries (mKdV) equation is examined. In particular, the collision between solitary waves with sech-type and algebraic (which only exist on a non-zero mean level) profiles is considered. An asymptotic transformation, valid if the higher-order coefficients satisfy a certain algebraic relationship, is used to transform the higher-order mKdV equation to an integrable member of the mKdV hierarchy. The transformation is used to show that the higher-order collision is asymptotically elastic and to derive the higher-order phase shifts. Numerical simulations of both elastic and inelastic collisions are performed. For the example covered ...


Numerical Simulation Of Contaminant Flow In A Wool Scour Bowl., Tim Marchant Dec 2006

Numerical Simulation Of Contaminant Flow In A Wool Scour Bowl., Tim Marchant

Tim Marchant

Wool scouring is the process of washing dirty wool after shearing. Our model numerically simulates contaminant movement in a wool scour bowl using the advection–dispersion equation. This is the first wool scour model to give time-dependent results and to model the transport of contaminants within a single scour bowl. Our aim is to gain a better understanding of the operating parameters that will produce efficient scouring. Investigating the effects of varying the parameters reveals simple, interesting relationships that give insight into the dynamics of a scour bowl.


An Undular Bore Solution For The Higher-Order Korteweg-De Vries Equation, Tim Marchant Dec 2005

An Undular Bore Solution For The Higher-Order Korteweg-De Vries Equation, Tim Marchant

Tim Marchant

Undular bores describe the evolution and smoothing out of an initial step in mean height and are frequently observed in both oceanographic and meteorological applications. The undular bore solution for the higher-order Korteweg-de Vries (KdV) equation is derived, using an asymptotic transformation which relates the KdV equation and its higher-order counterpart. The higher-order KdV equation considered includes all possible third-order correction terms (where the KdV equation retains second-order terms). The asymptotic transformation is then applied to the KdV undular bore solution to obtain the higher-order undular bore. Examples of higher-order undular bores, describing both surface and internal waves, are presented ...


Solitary Wave Interaction And Evolution For A Higher-Order Hirota Equation, Tim Marchant Dec 2005

Solitary Wave Interaction And Evolution For A Higher-Order Hirota Equation, Tim Marchant

Tim Marchant

Solitary wave interaction and evolution for a higher-order Hirota equation is examined. The higher-order Hirota equation is asymptotically transformed to a higher-order member of the NLS hierarchy of integrable equations, if the higher-order coefficients satisfy a certain algebraic relationship. The transformation is used to derive higher-order one- and two-soliton solutions. It is shown that the interaction is asymptotically elastic and the higher-order corrections to the coordinate and phase shifts are derived. For the higher-order Hirota equation resonance occurs between the solitary waves and linear radiation, so soliton perturbation theory is used to determine the details of the evolving wave and ...


Modelling A Wool Scour Bowl, Tim Marchant Dec 2005

Modelling A Wool Scour Bowl, Tim Marchant

Tim Marchant

Wool scouring is the process of washing dirty wool after shearing. Our model simulates, using the advection-diffusion equation, the movement of contaminants within a scour bowl. The effects of varying the important parameters are investigated. Interesting, but simple, relationships are found which give insight into the dynamics of a scour bowl.


Approximate Solutions For Magmon Propagation From A Reservoir, Tim Marchant Dec 2004

Approximate Solutions For Magmon Propagation From A Reservoir, Tim Marchant

Tim Marchant

A 1D partial differential equation (pde) describing the flow of magma in the Earth's mantle is considered, this equation allowing for compaction and distension of the surrounding matrix due to the magma. The equation has periodic travelling wave solutions, one limit of which is a solitary wave, called a magmon. Modulation equations for the magma equation are derived and are found to be either hyperbolic or of mixed hyperbolic/elliptic type, depending on the specific values of the wave number, mean height and amplitude of the underlying modulated wave. The periodic wave train is stable in the hyperbolic case ...


Microwave Thawing Of Cylinders., Tim Marchant Dec 2003

Microwave Thawing Of Cylinders., Tim Marchant

Tim Marchant

Microwave thawing of a cylinder is examined. The electromagnetic field is governed by Maxwell's equations, where the electrical conductivity and the thermal absorptivity are both assumed to depend on temperature. The forced heat equation governs the absorption and diffusion of heat where convective heating occurs at the surface of the cylinder, while the Stefan condition governs the position of the moving phase boundary. A semi-analytical model, which consists of ordinary differential equations, is developed using the Galerkin method. Semi-analytical solutions are found for the temperature, the electric-field amplitude in the cylinder and the position of the moving boundary. Two ...


Asymptotic Solitons For A Third-Order Kortweg-De Vries Equation, Tim Marchant Dec 2003

Asymptotic Solitons For A Third-Order Kortweg-De Vries Equation, Tim Marchant

Tim Marchant

Solitary wave interaction for a higher-order version of the Korteweg–de Vries (KdV) equation is considered. The equation is obtained by retaining third-order terms in the perturbation expansion, where for the KdV equation only first-order terms are retained. The third-order KdV equation can be asymptotically transformed to the KdV equation, if the third-order coefficients satisfy a certain algebraic relationship. The third-order two-soliton solution is derived using the transformation. The third-order phase shift corrections are found and it is shown that the collision is asymptotically elastic. The interaction of two third-order solitary waves is also considered numerically. Examples of an elastic ...


Semi-Analytical Solutions For One - And Two-Dimensional Pellet Problems., Tim Marchant Dec 2003

Semi-Analytical Solutions For One - And Two-Dimensional Pellet Problems., Tim Marchant

Tim Marchant

The problem of heat and mass transfer within a porous catalytic pellet in which an irreversible first–order exothermic reaction occurs is a much–studied problem in chemical–reactor engineering. The system is described by two coupled reaction–diffusion equations for the temperature and the degree of reactant conversion. The Galerkin method is used to obtain a semi–analytical model for the pellet problem with both one– and two–dimensional slab geometries. This involves approximating the spatial structure of the temperature and reactant–conversion profiles in the pellet using trial functions. The semi–analytical model is obtained by averaging the ...


Cubic Autocatalysis With Michaelis - Menten Kinetics: Semi-Analytical Solutions For The Reaction - Diffusion Cell, Tim Marchant Dec 2003

Cubic Autocatalysis With Michaelis - Menten Kinetics: Semi-Analytical Solutions For The Reaction - Diffusion Cell, Tim Marchant

Tim Marchant

Cubic-autocatalysis with Michaelis–Menten decay is considered in a one-dimensional reaction–diffusion cell. The boundaries of the reactor allow diffusion into the cell from external reservoirs, which contain fixed concentrations of the reactant and catalyst. The Galerkin method is used to obtain a semi-analytical model consisting of ordinary differential equations. This involves using trial functions to approximate the spatial structure of the reactant and autocatalyst concentrations in the reactor. The semi-analytical model is then obtained from the governing partial differential equations by averaging. The semi-analytical model allows steady-state concentration profiles and bifurcation diagrams to be obtained as the solution to ...


A Comparison Of Semi-Analytical And Numerical Solutions For The Microwave Heating Of A Lossy Material In A Three-Dimensional Waveguide, Prof. Tim Marchant Dec 2002

A Comparison Of Semi-Analytical And Numerical Solutions For The Microwave Heating Of A Lossy Material In A Three-Dimensional Waveguide, Prof. Tim Marchant

Tim Marchant

The microwave heating of a three-dimensional block in an infinitely long rectangular waveguide propagating the TE10 mode is considered. The electrical conductivity (the dielectric loss) is assumed to be a function of temperature, and modelled by the Arrhenius law. A coupled set of equations is obtained that describes the electromagnetic fields and the temperature distribution in the block. The numerical solutions of this problem are obtained by two methods, the well known FD-TD scheme and a frequency domain method which makes the further assumption that a single TE10 mode exists in the waveguide and material. The results show that an ...


Asymptotic Solitons For A Higher-Order Modified Korteweg–De Vries Equation, T. Marchant Oct 2002

Asymptotic Solitons For A Higher-Order Modified Korteweg–De Vries Equation, T. Marchant

Tim Marchant

Solitary wave interaction for a higher-order modified Korteweg–de Vries (mKdV) equation is examined. The higher-order mKdV equation can be asymptotically transformed to the mKdV equation, if the higher-order coefficients satisfy a certain algebraic relationship. The transformation is used to derive the higher-order two-soliton solution and it is shown that the interaction is asymptotically elastic. Moreover, the higher-order phase shifts are derived using the asymptotic theory. Numerical simulations of the interaction of two higher-order solitary waves are also performed. Two examples are considered, one satisfies the algebraic relationship derived from the asymptotic theory, and the other does not. For the ...


High-Order Interaction Of Solitary Waves On Shallow Water, Prof. Tim Marchant Dec 2001

High-Order Interaction Of Solitary Waves On Shallow Water, Prof. Tim Marchant

Tim Marchant

The interaction of solitary waves on shallow water is examined to fourth order. At first order the interaction is governed by the Korteweg-de Vries (KdV) equation, and it is shown that the unidirectional assumption, of right-moving waves only, is incompatible with mass conservation at third order. To resolve this, a mass conserving system of KdV equations, involving both right- and left-moving waves, is derived to third order. A fourth-order interaction term, in which the right- and left-moving waves are coupled, is also derived as this term is crucial in determining the fourth-order change in solitary wave amplitude. The form of ...


The Occurrence Of Limit-Cycles During Feedback Control Of Microwave Heating, Prof. Tim Marchant Dec 2001

The Occurrence Of Limit-Cycles During Feedback Control Of Microwave Heating, Prof. Tim Marchant

Tim Marchant

The microwave heating of one- and two-dimensional slabs, subject to linear feedback control, is examined. A semianalytical model of the microwave heating is developed using the Galerkin method. A local stability analysis of the model indicates that Hopf bifurcations occur; the regions of parameter space in which limit-cycles exist are identified. An efficient numerical scheme for the solution of the governing equations, which consist of the forced heat equation and a Helmholtz equation describing the electric-field amplitude, is also developed. An excellent comparison between numerical solutions of the semianalytical model and the governing equations is obtained for the temporal evolution ...


Cubic Autocatalytic Reaction-Diffusion Equations: Semi-Analytical Solutions, Prof. Tim Marchant Dec 2001

Cubic Autocatalytic Reaction-Diffusion Equations: Semi-Analytical Solutions, Prof. Tim Marchant

Tim Marchant

The Gray-Scott model of cubic-autocatalysis with linear decay is coupled with diffusion and considered in a one-dimensional reactor (a reaction-diffusion cell). The boundaries of the reactor are permeable, so diffusion occurs from external reservoirs that contain fixed concentrations of the reactant and catalyst. The Galerkin method is used to approximate the spatial structure of the reactant and autocatalyst concentrations in the reactor. Ordinary differential equations are then obtained as an approximation to the governing partial differential equations. The ordinary differential equations are then analysed to obtain semi-analytical results for the reaction-diffusion cell. Steady-state concentration profiles and bifurcation diagrams are obtained ...