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 Keyword

 Solitons and modulation theory (14)
 Nonlinear optics (7)
 Combustion theory (6)
 Microwave heating (4)
 Mathematical modelling (3)

 EQUATION (2)
 NLS equation; solitary waves; asymptotic transformation; elastic and inelastic collisions; higherorder phase and coordinate shifts (1)
 KdV equation; modulation theory; initial boundaryvalue problems (1)
 Reactiondiffusion equations; GrayScott model; singularity theory; Hopf bifurcations; semianalytical solutions (1)
 Reactiondiffusion equations catalytic pellet singularity theory Hopf bifurcations semianalytical solutions Arrhenius law (1)
 Higherorder Hirota equation; Asymptotic transformation; Solitary wave interaction; Higherorder coordinate and phase shifts; Embedded solitons; Soliton perturbation theory; Solitary wave tails (1)
 Combustion; Composting; Heat conduction; Nonlinear dynamics; Safety; Selfheating (1)
 Microwave heating; thermal runaway; Arrhenius law; TM waveguide mode; threedimensional bodies (1)
 Modified Kortewegde Vries equation; Soliton; Undular bore; Modulation theory; Initialboundary value problem (1)
 LIQUIDCRYSTALS (1)
 NEMATIC LIQUIDCRYSTALS; INTERNAL OSCILLATIONS; VARIATIONAL APPROACH; VECTOR SOLITONS; OPTICAL FIBERS; PROPAGATION (1)
 Liquid crystal; Soliton; Soliton dipole; Nematicon; Modulation theory (1)
 Reaction–diffusion equations; Cubic autocatalysis; Michaelis–Menten kinetics; Singularity theory; Hopf bifurcations; Semianalytical solutions (1)
 Microwave heating; Thawing; Stefan problem; Maxwell's equations; Semianalytical solutions (1)
 Reaction–diffusion–convection equations; Cubic autocatalysis; Applied electric field; Ionic migration; Singularity theory; Hopf bifurcations; Semianalytical solutions (1)
 Liquids and polymers Mathematical physics Computational physics Optics (1)
 Higher order Hirota equation; Embedded solitons; Soliton perturbation theory; Solitary wave tails (1)
 MODEL; RADIATION (1)
 NONLINEAR SCHRODINGEREQUATION; SOLITARY WAVE INTERACTION; DEEPWATER (1)
 RESONANT FLOW; TOPOGRAPHY; SOLITONS; WAVES (1)
 Soft matter (1)
 EQUATIONS (1)
 Initialboundary value problem; magma flow; modulation theory; undular bore; solitary waves (1)
 Quantum optics and lasers (1)
 Microwave heating; Limitcycles; Stability analysis; Hopf bifurcations; Control process; Thermal runaway (1)
Articles 1  30 of 34
FullText Articles in Physics
Reorientational Versus Kerr Dark And Gray Solitary Waves Using Modulation Theory, Prof. Tim Marchant
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 selfdefocusing 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
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 secondorder temporal and spatial discretization errors, are considered. These are the CrankNicolson scheme and a scheme, due to Taha [1], based on the inverse scattering transform. The firstorder 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
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 firstorder 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 HigherOrder Hirota Equation, Tim Marchant
Soliton Perturbation Theory For A HigherOrder 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 leftmoving waves. Also, a twoparameter 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
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 boundaryinduced bouncing of a nematicon in a onedimensional cell and it is found to give predictions in very good agreement with numerical solutions. The boundaryinduced motion is then considered numerically for a twodimensional 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 RunUps Around Islands, Tim Marchant
A Perturbation Drbem Model For Weakly Nonlinear Wave RunUps 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 runups of weakly nonlinear long waves scattered by islands. Under the assumption that the incident waves are harmonic, the timedependent nonlinear Boussinesq equations are transformed into three timeindependent linear equations by using the perturbation method, where, besides nonlinearity ε, the dispersion μ2 is also included in the perturbed expansion. The firstorder solution η0 is found by using the linear longwave equations. Then η0 is used in two secondorder governing equations related to the dispersion and nonlinearity, respectively. Since no any omission ...
Evolution Of HigherOrder Gray Hirota Solitary Waves, Tim Marchant
Evolution Of HigherOrder 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 higherorder defocusing Hirota equation is examined. A direct analysis is used to identify families of higherorder gray Hirota solitary waves, which are embedded for certain parameter values. Soliton perturbation theory is used to detmine the detailed behavior of an evolving higherorder gray Hirota solitary wave. An integral expression for the firstorder correction to the wave is found and analytical expressions for the steadystate and transient components of the solitary ...
Mathematical Modelling Of Nematicons And Their Interactions, Prof. Tim Marchant
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 InitialBoundary Value Problem For The Modified KortewegDe Vries Equation, Tim Marchant
Undular Bores And The InitialBoundary Value Problem For The Modified KortewegDe Vries Equation, Tim Marchant
Tim Marchant
Two types of analytical undular bore solutions, of the initial value problem for the modified Kortewegde 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 initialboundary value ...
Collisionless Shock Resolution In Nematic Liquid Crystals, Tim Marchant
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 selffocusing, 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 ...
SemiAnalytical Solutions For A GrayScott ReactionDiffusion Cell With An Applied Electric Field, Tim Marchant
SemiAnalytical Solutions For A GrayScott ReactionDiffusion 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 semianalytical 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 OneDimensional Nematicon Solution, Tim Marchant
Nonlocal Validity Of An Asymptotic OneDimensional 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 NonLocal Limit, Tim Marchant
Dipole Soliton Formation In A Nematic Liquid Crystal In The NonLocal 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 ...
SelfHeating In Compost Piles Due To Biological Effects, Tim Marchant
SelfHeating In Compost Piles Due To Biological Effects, Tim Marchant
Tim Marchant
The increase in temperature in compost piles/landfill sites due to microorganisms undergoing exothermic reactions is modelled. A simplified model is considered in which only biological selfheating 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 microorganisms die or become dormant at high temperatures. The bifurcation behaviour is investigated for 1d slab and 2d rectangular slab geometries. In both cases there ...
Solitary Wave Interaction For A HigherOrder Nonlinear Schrodinger Equation, Tim Marchant
Solitary Wave Interaction For A HigherOrder Nonlinear Schrodinger Equation, Tim Marchant
Tim Marchant
Solitary wave interaction for a higherorder version of the nonlinear Schrödinger (NLS) equation is examined. An asymptotic transformation is used to transform a higherorder NLS equation to a higherorder member of the NLS integrable hierarchy, if an algebraic relationship between the higherorder coefficients is satisfied. The transformation is used to derive the higherorder one and twosoliton solutions; in general, the Nsoliton solution can be derived. It is shown that the higherorder collision is asymptotically elastic and analytical expressions are found for the higherorder phase and coordinate shifts. Numerical simulations of the interaction of two higherorder solitary waves are also performed ...
Asymptotic Solitons On A NonZero Mean Level., Tim Marchant
Asymptotic Solitons On A NonZero Mean Level., Tim Marchant
Tim Marchant
The collision of solitary waves for a higherorder modified Kortewegde Vries (mKdV) equation is examined. In particular, the collision between solitary waves with sechtype and algebraic (which only exist on a nonzero mean level) profiles is considered. An asymptotic transformation, valid if the higherorder coefficients satisfy a certain algebraic relationship, is used to transform the higherorder mKdV equation to an integrable member of the mKdV hierarchy. The transformation is used to show that the higherorder collision is asymptotically elastic and to derive the higherorder 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
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 timedependent 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 HigherOrder KortewegDe Vries Equation, Tim Marchant
An Undular Bore Solution For The HigherOrder KortewegDe 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 higherorder Kortewegde Vries (KdV) equation is derived, using an asymptotic transformation which relates the KdV equation and its higherorder counterpart. The higherorder KdV equation considered includes all possible thirdorder correction terms (where the KdV equation retains secondorder terms). The asymptotic transformation is then applied to the KdV undular bore solution to obtain the higherorder undular bore. Examples of higherorder undular bores, describing both surface and internal waves, are presented ...
Solitary Wave Interaction And Evolution For A HigherOrder Hirota Equation, Tim Marchant
Solitary Wave Interaction And Evolution For A HigherOrder Hirota Equation, Tim Marchant
Tim Marchant
Solitary wave interaction and evolution for a higherorder Hirota equation is examined. The higherorder Hirota equation is asymptotically transformed to a higherorder member of the NLS hierarchy of integrable equations, if the higherorder coefficients satisfy a certain algebraic relationship. The transformation is used to derive higherorder one and twosoliton solutions. It is shown that the interaction is asymptotically elastic and the higherorder corrections to the coordinate and phase shifts are derived. For the higherorder 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
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 advectiondiffusion 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
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
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 semianalytical model, which consists of ordinary differential equations, is developed using the Galerkin method. Semianalytical solutions are found for the temperature, the electricfield amplitude in the cylinder and the position of the moving boundary. Two ...
Asymptotic Solitons For A ThirdOrder KortwegDe Vries Equation, Tim Marchant
Asymptotic Solitons For A ThirdOrder KortwegDe Vries Equation, Tim Marchant
Tim Marchant
Solitary wave interaction for a higherorder version of the Korteweg–de Vries (KdV) equation is considered. The equation is obtained by retaining thirdorder terms in the perturbation expansion, where for the KdV equation only firstorder terms are retained. The thirdorder KdV equation can be asymptotically transformed to the KdV equation, if the thirdorder coefficients satisfy a certain algebraic relationship. The thirdorder twosoliton solution is derived using the transformation. The thirdorder phase shift corrections are found and it is shown that the collision is asymptotically elastic. The interaction of two thirdorder solitary waves is also considered numerically. Examples of an elastic ...
SemiAnalytical Solutions For One  And TwoDimensional Pellet Problems., Tim Marchant
SemiAnalytical Solutions For One  And TwoDimensional 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: SemiAnalytical Solutions For The Reaction  Diffusion Cell, Tim Marchant
Cubic Autocatalysis With Michaelis  Menten Kinetics: SemiAnalytical Solutions For The Reaction  Diffusion Cell, Tim Marchant
Tim Marchant
Cubicautocatalysis with Michaelis–Menten decay is considered in a onedimensional 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 semianalytical 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 semianalytical model is then obtained from the governing partial differential equations by averaging. The semianalytical model allows steadystate concentration profiles and bifurcation diagrams to be obtained as the solution to ...
A Comparison Of SemiAnalytical And Numerical Solutions For The Microwave Heating Of A Lossy Material In A ThreeDimensional Waveguide, Prof. Tim Marchant
A Comparison Of SemiAnalytical And Numerical Solutions For The Microwave Heating Of A Lossy Material In A ThreeDimensional Waveguide, Prof. Tim Marchant
Tim Marchant
The microwave heating of a threedimensional 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 FDTD 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 HigherOrder Modified Korteweg–De Vries Equation, T. Marchant
Asymptotic Solitons For A HigherOrder Modified Korteweg–De Vries Equation, T. Marchant
Tim Marchant
Solitary wave interaction for a higherorder modified Korteweg–de Vries (mKdV) equation is examined. The higherorder mKdV equation can be asymptotically transformed to the mKdV equation, if the higherorder coefficients satisfy a certain algebraic relationship. The transformation is used to derive the higherorder twosoliton solution and it is shown that the interaction is asymptotically elastic. Moreover, the higherorder phase shifts are derived using the asymptotic theory. Numerical simulations of the interaction of two higherorder 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 ...
HighOrder Interaction Of Solitary Waves On Shallow Water, Prof. Tim Marchant
HighOrder 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 Kortewegde Vries (KdV) equation, and it is shown that the unidirectional assumption, of rightmoving waves only, is incompatible with mass conservation at third order. To resolve this, a mass conserving system of KdV equations, involving both right and leftmoving waves, is derived to third order. A fourthorder interaction term, in which the right and leftmoving waves are coupled, is also derived as this term is crucial in determining the fourthorder change in solitary wave amplitude. The form of ...
The Occurrence Of LimitCycles During Feedback Control Of Microwave Heating, Prof. Tim Marchant
The Occurrence Of LimitCycles During Feedback Control Of Microwave Heating, Prof. Tim Marchant
Tim Marchant
The microwave heating of one and twodimensional 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 limitcycles 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 electricfield 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 ReactionDiffusion Equations: SemiAnalytical Solutions, Prof. Tim Marchant
Cubic Autocatalytic ReactionDiffusion Equations: SemiAnalytical Solutions, Prof. Tim Marchant
Tim Marchant
The GrayScott model of cubicautocatalysis with linear decay is coupled with diffusion and considered in a onedimensional reactor (a reactiondiffusion 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 semianalytical results for the reactiondiffusion cell. Steadystate concentration profiles and bifurcation diagrams are obtained ...