Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 2001"
TL;DR: The Camassa-Holm equation as mentioned in this paper has a number of constants of motion arising as eigenvalues of an associated spectral problem, and the spectral picture is described and discussed in detail.
Abstract: The Camassa–Holm equation has a number of constants of motion arising as eigenvalues of an associated spectral problem. We give a description of the spectral picture and discuss the scattering problem.
671 citations
TL;DR: In this article, the authors derived the equation for a single species population with two age classes and a fixed maturation period living in a spatially unbounded environment, and showed that if the mature dea...
Abstract: In this paper, we derive the equation for a single species population with two age classes and a fixed maturation period living in a spatially unbounded environment. We show that if the mature deat...
305 citations
TL;DR: In this paper, the authors explore the geometry of complex vector waves by regarding them as a field of polarization ellipses, where singularities of this field are the C lines and L lines, where the polarization is purely circular and purely linear, respectively.
Abstract: Following Nye & Hajnal, we explore the geometry of complex vector waves by regarding them as a field of polarization ellipses. Singularities of this field are the C lines and L lines, where the polarization is purely circular and purely linear, respectively. The singularities can be reinterpreted as loci of photon spin 1 (C lines) and 0 (L lines). For Gaussian random superpositions of plane waves equidistributed in direction but with an arbitrary frequency spectrum, we calculate the density (length per unit volume) of C and L lines.
298 citations
TL;DR: In this article, exact solutions of the Helmholtz equation are constructed, possessing wavefront dislocation lines (phase singularities) in the form of knots or links where the wave function vanishes (knotted nothings) by making a nongeneric structure with a strength n dislocation loop threaded by a strength m dislocation line, and perturbing this.
Abstract: Exact solutions of the Helmholtz equation are constructed, possessing wavefront dislocation lines (phase singularities) in the form of knots or links where the wave function vanishes (‘knotted nothings’). The construction proceeds by making a nongeneric structure with a strength n dislocation loop threaded by a strength m dislocation line, and then perturbing this. In the resulting unfolded (stable) structure, the dislocation loop becomes an ( m , n ) torus knot if m and n are coprime, and N linked rings or knots if m and n have a common factor N ; the loop or rings are threaded by an m -stranded helix. In our explicit implementation, the wave is a superposition of Bessel beams, accessible to experiment. Paraxially, the construction fails.
207 citations
TL;DR: In this paper, the origin of the non-radially symmetric solutions for the Liouville problem was clarified by Chanillo and Kiessling, who showed that the solutions always inherit the invariance of the problem under inversion with respect to suitable circles.
Abstract: In the plane R2, we classify all solutions for an elliptic problem of Liouville type involving a (radial) weight function. As a consequence, we clarify the origin of the non-radially symmetric solutions for the given problem, as established by Chanillo and Kiessling.For a more general class of Liouville-type problems, we show that, rather than radial symmetry, the solutions always inherit the invariance of the problem under inversion with respect to suitable circles. This symmetry result is derived with the help of a 'shrinking-sphere' method.
182 citations
TL;DR: In this article, the directionality of the wave field has a profound effect upon the nonlinearity of a large wave event, and it is shown that a large number of waves, of varying frequency and propagating in different directions, were focused at one point in space and time to produce a large transient wave group.
Abstract: This paper describes a new laboratory study in which a large number of waves, of varying frequency and propagating in different directions, were focused at one point in space and time to produce a large transient wave group. A focusing event of this type is believed to be representative of the evolution of an extreme ocean wave in deep water. Measurements of the water–surface elevation and the underlying water–particle kinematics are compared with both a linear solution and a second–order solution based on the sum of the interactions first identified by Longuet–Higgins & Stewart. Comparisons between these data confirm that the directionality of the wavefield has a profound effect upon the nonlinearity of a large wave event. If the sum of the wave amplitudes generated at the wave paddles is held constant, an increase in the directional spread of the wavefield leads to lower maximum crest elevations. Conversely, if the generated wave amplitudes are increased until the onset of wave breaking, at or near the focal position, an increase in the directional spread allows larger limiting waves to evolve. An explanation of these results lies in the redistribution of the wave energy within the frequency domain. In the most nonlinear wave cases, neither the water–surface elevation nor the water–particle kinematics can be explained in terms of the free waves generated at the wave paddles and their associated bound waves. Indeed, the laboratory data suggest that there is a rapid widening of the free–wave regime in the vicinity of a large wave event. For a constant input–amplitude sum, these important spectral changes are shown to be strongly dependent upon the directionality of the wavefield. These findings explain the very large water–surface elevations recorded in previous unidirectional wave studies and the apparent contrast between unidirectional results and recent field data in which large directionally spread waves were shown to be much less nonlinear. The present study clearly demonstrates the need to incorporate the directionality of a wavefield if extreme ocean waves are to be accurately modelled and their physical characteristics explained.
149 citations
TL;DR: In this paper, Stokes developed a general constitutive relation which admitted the possibility that the viscosity could depend on the pressure, and this assumption was used in the seminal paper on fluid motion.
Abstract: In his seminal paper on fluid motion, Stokes developed a general constitutive relation which admitted the possibility that the viscosity could depend on the pressure. Such an assumption is particul...
140 citations
TL;DR: It is demonstrated that entropically stabilized lattice Boltzmann algorithms, while fully explicit and perfectly conservative, may achieve remarkably low values for transport coefficients, such as viscosity, and holds promise for high‐Reynolds‐number simulations of the Navier‐Stokes equations.
Abstract: We present a general methodology for constructing lattice Boltzmann models of hydrodynamics with certain desired features of statistical physics and kinetic theory. We show how a methodology of lin...
138 citations
TL;DR: In this article, a compactness theorem was proved for the eikonal equation in the plane, and it was shown that if fE (A )g° #0 is uniformly bounded, then frA g° # 0 is compact in L.
Abstract: in the plane. As ° ! 0, this functional favours jrAj = 1 and penalizes singularities where jrrAj concentrates. Our main result is a compactness theorem: if fE ° (A ° )g° #0 is uniformly bounded, then frA ° g° #0 is compact in L . Thus, in the limit ° ! 0, A solves the eikonal equation jrAj= 1 almost everywhere. Our analysis uses entropy relations’ and the div-curl lemma,’ adopting Tartar’ s approach to the interaction of linear di® erential equations and nonlinear algebraic relations.
137 citations
TL;DR: In this paper, a method for solving boundary value problems for linear partial differential equations (PDEs) in convex polygons is introduced. But the method is based on the existence of a simple global relation formulated in the complex k-plane, and on the invariant properties of this relation.
Abstract: A method is introduced for solving boundary‐value problems for linear partial differential equations (PDEs) in convex polygons. It consists of three algorithmic steps.
(1) Given a PDE , construct two compatible eigenvalue equations.
(2) Given a polygon , perform the simultaneous spectral analysis of these two equations. This yields an integral representation in the complex k ‐plane of the solution q (x1,x2) in terms of a function q ( k ), and an integral representation in the (x1, x2)‐plane of q( k ) in terms of the values of q and of its derivatives on the boundary of the polygon. These boundary values are in general related, thus only some of them can be prescribed.
(3) Given appropriate boundary conditions , express the part of q ( k ) involving the unknown boundary values in terms of the boundary conditions. This is based on the existence of a simple global relation formulated in the complex k ‐plane, and on the invariant properties of this relation.
As an illustration, the following integral representations are obtained: (a) q (x, t ) for a general dispersive evolution equation of order n in a domain bounded by a linearly moving boundary; (b) q (x,y) for the Laplace, modified Helmholtz and Helmholtz equations in a convex polygon. These general formulae and the analysis of the associated global relations are used to discuss typical boundary‐value problems for evolution equations and for elliptic equations.
129 citations
TL;DR: In this article, a new approach for perturbative calculation of Dirichlet{Neumann operators (DNOs) on domains that are perturbations of simple geometries is presented.
Abstract: This paper outlines the theoretical background of a new approach towards an accurate and well-conditioned perturbative calculation of Dirichlet{Neumann operators (DNOs) on domains that are perturbations of simple geometries. Previous work on the analyticity of DNOs has produced formulae that, as we have found, are very ill-conditioned. We show how a simple change of variables can lead to recursions that satisfy analyticity estimates without relying on subtle cancellation properties at the heart of previous formulae.
TL;DR: In this paper, a nonlinear stability analysis is performed for the Darcy equations of thermal convection in a fluid-saturated porous medium when the medium is rotating about an axis orthogonal to the layer in the direction of gravity.
Abstract: A nonlinear stability analysis is performed for the Darcy equations of thermal convection in a fluid‐saturated porous medium when the medium is rotating about an axis orthogonal to the layer in the direction of gravity. A best possible result is established in that we show that the global nonlinear stability Rayleigh number is exactly the same as that for linear instability. It is important to realize that the nonlinear stability boundary holds unconditionally, i.e. for all initial data, and thus for the rotating porous convection problem governed by the Darcy equations, subcritical instabilities are not possible.
TL;DR: In this article, the structure of the random attractor for the Chafee-Infante reaction-diffusion equation perturbed by a multiplicative white noise was studied in an infinite-dimensional setting.
Abstract: We study in some detail the structure of the random attractor for the Chafee-Infante reaction-diffusion equation perturbed by a multiplicative white noise, d u = ( Δ u + β u - u 3 ) d t + σ u o d W t , x ∈ D ⊂ R m First we prove, for m ⩽ 5, a lower bound on the dimension of the random attractor, which is of the same order in β as the upper bound we derived in an earlier paper, and is the same as that obtained in the deterministic case. Then we show, for m = 1, that as β passes through λ 1 (the first eigenvalue of the negative Laplacian) from below, the system undergoes a stochastic bifurcation of pitchfork type. We believe that this is the first example of such a stochastic bifurcation in an infinite–dimensional setting. Central to our approach is the existence of a random unstable manifold.
TL;DR: In this article, the existence and uniqueness of solutions to Navier and Stokes equations with hereditary characteristics were proved. But they did not consider the case when the external force contains some hereditary characteristics.
Abstract: Some results on the existence and uniqueness of solutions to Navier;Stokes equations when the external force contains some hereditary characteristics are proved.
TL;DR: The violent bubble collapses lead to temperatures of several thousand kelvin, which drive chemical activity as mentioned in this paper, and can lead to the formation of new cavitation bubbles to increase chemical activity.
Abstract: Sonochemistry involves focusing acoustic energy through cavitation bubbles to increase chemical activity. The violent bubble collapses lead to temperatures of several thousand kelvin, which drive c...
TL;DR: In this paper, a predator-prey system with Holling-Tanner interaction terms was studied, and it was shown that if the saturation rate m is large, spatially inhomogeneous steady-state solutions arise, contrasting sharply with the small m case, where no such solution could exist.
Abstract: We study a predator–prey system with Holling–Tanner interaction terms. We show that if the saturation rate m is large, spatially inhomogeneous steady-state solutions arise, contrasting sharply with the small-m case, where no such solution could exist. Furthermore, for large m, we give sharp estimates on the ranges of other parameters where spatially inhomogeneous solutions can exist. We also determine the asymptotic behaviour of the spatially inhomogeneous solutions as m → ∞, and an interesting relation between this population model and free boundary problems is revealed.
TL;DR: Hencky's strain-energy function for finite isotropic elasticity was obtained by the replacement of the infinitesimal strain measure occurring in the classical strain energy function.
Abstract: Hencky's strain-energy function for finite isotropic elasticity is obtained by the replacement of the infinitesimal strain measure occurring in the classical strain-energy function of infinitesimal...
TL;DR: In this paper, a mesh termination scheme in acoustic scattering, known as the perfectly matched layer (PML) method, was studied, where the scatterer is contained in a bounded and strictly convex artificial domain.
Abstract: In this work, we study a mesh termination scheme in acoustic scattering, known as the perfectly matched layer (PML) method. The main result of the paper is the following. Assume that the scatterer is contained in a bounded and strictly convex artificial domain. We surround this domain by a PML of constant thickness. On the peripheral boundary of this layer, a homogenous Dirichlet condition is imposed. We show in this paper that the resulting boundary-value problem for the scattered field is uniquely solvable for all wavenumbers and the solution within the artificial domain converges exponentially fast toward the full-space scattering solution when the layer thickness is increased. The proof is based on the idea of interpreting the PML medium as a complex stretching of the coordinates in Rn and on the use of complexified layer potential techniques.
TL;DR: The present algorithm can evaluate accurately, on a personal computer, scattering from bodies of acoustical sizes of several hundreds and exhibits super–algebraic convergence, and it does not suffer from accuracy breakdowns of any kind.
Abstract: We present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in threedimensional space. This algorithm evaluates scattered fields through fast, highorder, ac...
TL;DR: A review of laser-triggered explosive devices can be found in this article, where a range of the experiments conducted and theories developed for this novel field are discussed, and the fundamental mechanisms remain to be fully explained.
Abstract: Energetic materials are conventionally ignited by the application of heat to some part of an explosive target. This is most often provided by a flame or by electrical heating using a resistive wire. The material responds by thermally heating and starting a burning zone, which spreads out from the ignition point generating gas. In some cases this zone can accelerate (due to the effect of this gas) into a reactive shock wave, termed detonation. Lasers are used in a variety of applications to thermal heat a range of materials, and thus seem an obvious candidate to trigger chemical reaction in energetic ones. A light pulse offers several advantages over an electrical one, since it may be delivered down a path both immune to electrical effects and chemically stable. Thus, the triggering of safety apparatus (such as the firing of bolts on aircraft exits) represents a major thrust in the development of laser–triggered explosive devices. The energy of the pulse may be used in one of two ways to achieve these effects. In the first, the laser is shone directly upon the chemical medium, which absorbs the light at discrete wavelengths. In the second, it is used to vaporize a metallic flyer that is launched to impact on a target creating a high–pressure zone. Either mechanism starts the required reaction. However, when the pulse is delivered directly to the material, detonation is found to proceed immediately with no transition via burning. This makes the process inexplicable using present concepts. This review will address a range of the experiments conducted and theories developed for this novel field. However, it should be emphasized that the fundamental mechanisms remain to be fully explained making the application both academically stimulating as well as industrially important.
TL;DR: In this article, the problem of neutral inclusions for two-dimensional conductivity (or equivalently, anti-plane elasticity) is considered, where the inclusion is assumed to have a hole (or perfect conductor) at its core surrounded by a thick coating of isotropic material.
Abstract: The problem of neutral inclusions for two-dimensional conductivity (or, equivalently, anti-plane elasticity) is considered. Such an inclusion when inserted in a matrix containing a uniform applied electric field does not disturb the field outside the inclusion. Consequently, assemblages of neutral inclusions have certain moduli of their effective conductivity tensor that can be determined exactly. The inclusion is assumed to have a hole (or perfect conductor) at its core surrounded by a thick coating of isotropic material. The whole construction is embedded in a possibly anisotropic matrix. Analytic formulae for the boundaries of the core and coating are found with the use of conformal mapping techniques. The admissible inclusion shapes depend on the applied electric field and on the conductivities of matrix and coating. It is shown that the inclusions can have a variety of shapes and are not just restricted to being coated confocal ellipses. However, coated confocal ellipses are the only inclusions which are neutral to multiple applied fields.
TL;DR: In this article, the spectral dependence of the degree and angle of polarization of skylight at 90° from the Sun along the antisolar meridian along the polar meridian is analyzed.
Abstract: Using 180° field–of–view (full–sky) imaging polarimetry, the patterns of the degree and angle of polarization of the entire summer sky were measured on 25 June 1999 at a location north of the Arctic Circle in Finnish Lapland as a function of the angular solar zenith distance. A detailed description of the used full–sky imaging polarimeter and its calibration is given. A series of the degree and angle of polarization pattern of the full sky is presented in the form of high–resolution circular maps measured in the blue (450 nm) spectral range as a function of the solar zenith distance. Graphs of the spectral dependence of the degree and angle of polarization of skylight at 90° from the Sun along the antisolar meridian are shown. The celestial regions of negative polarization and the consequence of the existence of this anomalous polarization, the neutral points, are visualized. The measured values of the angular zenith distance of the Arago and Babinet neutral points are presented as a function of the zenith distance of the Sun for the red (650 nm), green (550 nm) and blue (450 nm) ranges of the spectrum. The major aim of this work is to give a clear and comprehensive picture, with the help of full–sky imaging polarimetry, of what is going on in the entire polarized skydome. We demonstrate how variable the degree of polarization of skylight and the position of the neutral points can be within 24 h on a sunny, almost cloudless, visually clear day.
TL;DR: In this article, a rigorous existence theory for three-dimensional steady gravity-capillary finite-depth water waves which are uniformly translating in one horizontal spatial direction x and periodic in the transverse direction z is presented.
Abstract: This paper contains a rigorous existence theory for three-dimensional steady gravity-capillary finite-depth water waves which are uniformly translating in one horizontal spatial direction x and periodic in the transverse direction z. Physically motivated arguments are used to find a formulation of the problem as an infinite-dimensional Hamiltonian system in which x is the time-like variable, and a centre-manifold reduction technique is applied to demonstrate that the problem is locally equivalent to a finite-dimensional Hamiltonian system. General statements concerning the existence of waves which are periodic or quasiperiodic in x (and periodic in z) are made by applying standard tools in Hamiltonian-systems theory to the reduced equations.A critical curve in Bond number–Froude number parameter space is identified which is associated with bifurcations of generalized solitary waves. These waves are three dimensional but decay to two-dimensional periodic waves (small-amplitude Stokes waves) far upstream and downstream. Their existence as solutions of the water-wave problem confirms previous predictions made on the basis of model equations.
TL;DR: In this article, the authors studied dielectric breakdown for composites made of two isotropic phases and showed that Sachs's bound is optimal for the case of isotropics.
Abstract: We study dielectric breakdown for composites made of two isotropic phases. We show that Sachs's bound is optimal. This simple example is used to illustrate a variational principle which departs from the traditional one. We also derive the usual variational principle by elementary means without appealing to the technology of convex duality.
TL;DR: For a Helmholtz eigenvalue problem with a multiply-connected domain, the boundary integral equation approach as well as the boundary element method is shown to yield spurious eigenvalues even if the complex-valued kernel is used as mentioned in this paper.
Abstract: For a Helmholtz eigenvalue problem with a multiply connected domain, the boundary integral equation approach as well as the boundary-element method is shown to yield spurious eigenvalues even if the complex-valued kernel is used In such a case, it is found that spurious eigenvalues depend on the geometry of the inner boundary Demonstrated as an analytical case, the spurious eigenvalue for a multiply connected problem with its inner boundary as a circle is studied analytically By using the degenerate kernels and circulants, an annular case can be studied analytically in a discrete system and can be treated as a special case The proof for the general boundary instead of the circular boundary is also derived The Burton-Miller method is employed to eliminate spurious eigenvalues in the multiply connected case Moreover, a modified method considering only the real-part formulation is provided Five examples are shown to demonstrate that the spurious eigenvalues depend on the shape of the inner boundary Good agreement between analytical prediction and numerical results are found
TL;DR: In this article, the Cosserat director theory is used to formulate the problem of a long thin weightless rod constrained, by suitable distributed forces, to lie on a cylinder while being held by end tension and twisting moment.
Abstract: The Cosserat director theory is used to formulate the problem of a long thin weightless rod constrained, by suitable distributed forces, to lie on a cylinder while being held by end tension and twisting moment. Applications of this problem are found, for instance, in the buckling of drill strings inside a cylindrical hole. In the case of a rod of isotropic cross‐section the equilibrium equations can be reduced to those of a one‐degree‐of‐freedom oscillator in terms of the angle that the local tangent to the rod makes with the axis of the cylinder. Depending on the radius of the cylinder and the applied load, the oscillator has several fixed points, each of which corresponds to a different helical solution of the rod. More complicated shapes are also possible, and special attention is given to localized configurations described by homoclinic orbits of the oscillator. Heteroclinic saddle connections are found to play an important role in the post‐buckling behaviour by defining critical loads at which a straight rod may coil up into a helix.
TL;DR: In this paper, a rigorous procedure for the derivation of global watershedscale balance laws for mass, momentum, energy and entropy has been pursued, and a set of watershed scale balance laws are presented.
Abstract: In previous work by the authors a rigorous procedure for the derivation of global watershedscale balance laws for mass, momentum, energy and entropy has been pursued. To complement these, a set of ...
TL;DR: In this article, a view that there is practical value in being able to measure aspects of complexity in industrial, particularly manufacturing, systems is presented, at a decreasing level of complexity.
Abstract: This paper is motivated by a view that there is practical value in being able to measure aspects of complexity in industrial, particularly manufacturing, systems. We address, at a decreasing level ...
TL;DR: A simple nonlinear model is proposed aimed at illustrating the most salient features of the micromechanics of uniaxially stretched solid foams and shows that the stretch heterogeneity observed in experiments stems from the lack of convexity of the governing energy functional, which favours two characteristic values of local stretch.
Abstract: Compressed open–cell solid foams frequently exhibit spatially heterogeneous distributions of local stretch. The theoretical aspects of this deformation habit have not been clearly elucidated. Here we propose a simple nonlinear model aimed at illustrating the most salient features of the micromechanics of uniaxially stretched solid foams. Then we study the energetics of the model to show that the stretch heterogeneity observed in experiments stems from the lack of convexity of the governing energy functional, which favours two characteristic values of local stretch. These characteristic values are independent of the applied overall stretch and define two configurational phases of the foam. The predicted stretch distributions correspond to stratified mixtures of the phases; stretching occurs in the form of a phase transformation, by growth of one of the phases at the expense of the other. We also compare the predicted mechanical response with experimental data for a series of foams of different densities and discuss the analogy between the stretching of foams and the liquefaction of van der Waals gases. Lastly, we perform displacement field measurements using the digital image correlation technique and find the results to be in agreement with our predictions.
TL;DR: In this article, Green's functions and propagators for the multidimensional anisotropic spacetime fractional diffusion equation are obtained in integral form, and uniqueness of the solutions is studied in the framework.
Abstract: Green's functions and propagators for the multidimensional anisotropic spacetime fractional diffusion equation are obtained in integral form. Uniqueness of the solutions is studied in the framework...