Existence and Homogenisation of Travelling Waves Bifurcating from Resonances of Reaction–Diffusion Equations in Periodic Media

The existence of travelling wave type solutions is studied for a scalar reaction diffusion equation in \(\mathbb {R}^2\) with a nonlinearity which depends periodically on the spatial variable. We treat the coefficient of the linear term as a parameter and we formulate the problem as an infinite spatial dynamical system. Using a centre manifold reduction we obtain a finite dimensional dynamical system on the centre manifold with fully degenerate linear part. By phase space analysis and Conley index methods we find conditions on the parameter and nonlinearity for the existence of travelling wave type solutions with particular wave speeds. The analysis provides an approach to the homogenisation problem as the period of the periodic dependence in the nonlinearity tends to zero.     

Analysis of Newton’s Method to Compute Travelling Waves in Discrete Media

We present a variant of Newton’s method for computing travelling wave solutions to scalar bistable lattice differential equations. We prove that the method converges to a solution, obtain existence and uniqueness of solutions to such equations with a small second order term and study the limiting behaviour of such solutions as this second order term tends to zero. The robustness of the algorithm will be discussed using numerical examples. These results will also be used to illustrate phenomena like propagation failure, which are encountered when studying lattice differential equations. We finish by discussing the broad application range of the method and illustrate that higher dimensional systems exhibit richer behaviour than their scalar counterparts.     

New inroads in an old subject: Plasticity, from around the atomic to the macroscopic scale

Nonsingular, stressed, dislocation (wall) profiles are shown to be 1-d equilibria of a non-equilibrium theory of Field Dislocation Mechanics (FDM). It is also shown that such equilibrium profiles corresponding to a given level of load cannot generally serve as a travelling wave profile of the governing equation for other values of nearby constant load; however, one case of soft loading with a special form of the dislocation velocity law is demonstrated to have no ‘Peierls barrier’ in this sense. The analysis is facilitated by the formulation of a 1-d, scalar, time-dependent, Hamilton-Jacobi equation as an exact special case of the full 3-d FDM theory accounting for non-convex elastic energy, small, Nye-tensor-dependent core energy, and possibly an energy contribution based on incompatible slip. Relevant nonlinear stability questions, including that of nucleation, are formulated in a non-equilibrium setting. Elementary averaging ideas show a singular perturbation structure in the evolution of the (unsymmetric) macroscopic plastic distortion, thus pointing to the possibility of predicting generally rate-insensitive slow response constrained to a tensorial ‘yield’ surface, while allowing fast excursions off it, even though only simple kinetic assumptions are employed in the microscopic FDM theory. The emergent small viscosity on averaging that serves as the small parameter for the perturbation structure is a robust, almost-geometric consequence of large gradients of slip in the dislocation core and the persistent presence of a large number of dislocations in the averaging volume. In the simplest approximation, the macroscopic yield criterion displays anisotropy based on the microscopic dislocation line and Burgers vector distribution, a dependence on the Laplacian of the incompatible slip tensor and a nonlocal term related to a Stokes-Helmholtz-curl projection of an ‘internal stress’ derived from the incompatible slip energy.     

Kinetic Relations for a Lattice Model of Phase Transitions

The aim of this article is to analyse travelling waves for a lattice model of phase transitions, specifically the Fermi–Pasta–Ulam chain with piecewise quadratic interaction potential. First, for fixed, sufficiently large subsonic wave speeds, we rigorously prove the existence of a family of travelling wave solutions. Second, it is shown that this family of solutions gives rise to a kinetic relation which depends on the jump in the oscillatory energy in the solution tails. Third, our constructive approach provides a very good approximate travelling wave solution.     

Travelling Wave Solutions in Multigroup Age-Structured Epidemic Models

Age-structured epidemic models have been used to describe either the age of individuals or the age of infection of certain diseases and to determine how these characteristics affect the outcomes and consequences of epidemiological processes. Most results on age-structured epidemic models focus on the existence, uniqueness, and convergence to disease equilibria of solutions. In this paper we investigate the existence of travelling wave solutions in a deterministic age-structured model describing the circulation of a disease within a population of multigroups. Individuals of each group are able to move with a random walk which is modelled by the classical Fickian diffusion and are classified into two subclasses, susceptible and infective. A susceptible individual in a given group can be crisscross infected by direct contact with infective individuals of possibly any group. This process of transmission can depend upon the age of the disease of infected individuals. The goal of this paper is to provide sufficient conditions that ensure the existence of travelling wave solutions for the age-structured epidemic model. The case of two population groups is numerically investigated which applies to the crisscross transmission of feline immunodeficiency virus (FIV) and some sexual transmission diseases.     

Size effects and idealized dislocation microstructure at small scales: Predictions of a Phenomenological model of Mesoscopic Field Dislocation Mechanics: Part II

In Part I of this set of two papers, a model of mesoscopic plasticity is developed for studying initial-boundary value problems of small scale plasticity. Here we make qualitative, finite element method-based computational predictions of the theory. We demonstrate size effects and the development of strong inhomogeneity in simple shearing of plastically constrained grains. Non-locality in elastic straining leading to a strong Bauschinger effect is analyzed. Low shear strain boundary layers in constrained simple shearing of infinite layers of polycrystalline materials are not predicted by the model, and we justify the result based on an examination of the no-dislocation-flow boundary condition. The time-dependent, spatially homogeneous, simple shearing solution of PMFDM is studied numerically. The computational results and an analysis of continuous dependence with respect to initial data of solutions for a model linear problem point to the need for a nonlinear study of a stability transition of the homogeneous solution with decreasing grain size and increasing applied deformation. The continuous-dependence analysis also points to a possible mechanism for the development of spatial inhomogeneity in the initial stages of deformation in lower-order gradient plasticity theory. Results from thermal cycling of small scale beams/films with different degrees of constraint to plastic flow are presented showing size effects and reciprocal-film-thickness scaling of dislocation density boundary layer width. Qualitative similarities with results from discrete dislocation analyses are noted where possible.We discuss the convergence of approximate solutions with mesh refinement and its implications for the prediction of dislocation microstructure development, motivated by the notion of measure-valued solutions to conservation laws.     

Existence and Stability of a Screw Dislocation under Anti-Plane Deformation

We formulate a variational model for a geometrically necessary screw dislocation in an anti-plane lattice model at zero temperature. Invariance of the energy functional under lattice symmetries renders the problem non-coercive. Nevertheless, by establishing coercivity with respect to the elastic strain and a concentration compactness principle, we prove the existence of a global energy minimizer and thus demonstrate that dislocations are globally stable equilibria within our model.     

Slow motion of a slip spherical particle perpendicular to two plane walls

The problem of the quasisteady motion of a spherical fluid or solid particle with a slip-flow surface in a viscous fluid perpendicular to two parallel plane walls at an arbitrary position between them is investigated theoretically in the limit of small Reynolds number. To solve the axisymmetric Stokes equation for the fluid velocity field, a general solution is constructed from the superposition of the fundamental solutions in both circular cylindrical and spherical coordinate systems. The boundary conditions are enforced first at the plane walls by the Hankel transform and then on the particle surface by a collocation technique. Numerical results for the hydrodynamic drag force exerted on the particle are obtained with good convergence for various values of the relative viscosity or slip coefficient of the particle and of the relative separation distances between the particle and the confining walls. For the motions of a spherical particle normal to a single plane wall and of a no-slip sphere perpendicular to two plane walls, our drag results are in good agreement with the available solutions in the literature for all relative particle-to-wall spacings. The boundary-corrected drag force acting on the particle in general increases with an increase in its relative viscosity or with a decrease in its slip coefficient for a given geometry, but there are exceptions. For a specified wall-to-wall spacing, the drag force is minimal when the particle is situated midway between the two plane walls and increases monotonically when it approaches either of the walls. The boundary effect on the particle motion normal to two plane walls is found to be significant and much stronger than that parallel to them.     

Elastic precursor decay and radiation from nonuniformly moving dislocations

Precursor decay in plate impact experiments on single crystals is re-examined from the viewpoint of the elastodynamics of moving dislocations. Superposition of solutions for many dislocations set in motion by an incident plane wave is used to relate the decay of the wave amplitude at the front of the plane wave to the density and velocity of dislocations at the wavefront. The resulting precursor decay relation is the same as the one derived from an elastic/visco-plastic model of the material, except for a small correction due to differences between the effects of forward and backward propagating dislocations. Motivated by this added support for the validity of the precursor decay equation, the values used for the quantities in this equation are re-examined. Recent experimental results and the elastodynamics analysis are interpreted as indicating that the commonly-used values of dislocation velocity are probably satisfactory, but that the values used for dislocation density are several orders of magnitude too small near the lapped surfaces of the crystal. These large dislocation densities are identified as the probable dominant cause of the lower-than-predicted precursor amplitudes that are recorded in experiments. More accurate experimental data and inclusion of the non-linear elasticity effects are essential in determining whether or not the observed precursor decay in the bulk of the specimen can be explained by the motion of dislocations present initially. Calculations of energy radiated from screw and edge dislocations that start from rest and move thereafter at constant velocity confirm that dislocation drag forces due to continuum elasticity effects are small for dislocation velocities less than, say, 80% of the elastic shear wave speed. At supersonic speeds the continuum drag effects become so large that sustained supersonic motion of dislocations appears unlikely.     

Stability of travelling fronts in a model for flame propagation part I: Linear analysis

We investigate the stability of travelling wave solutions of the multidimensional thermodiffusive model for flame propagation with unit Lewis number. This model consists in a system of two nonlinear parabolic equations posed in an infinite cylinder, with Neumann boundary conditions. In this paper, we prove that every travelling wave solution of this model is linearly stable. Our tools are exponential decay estimates for solutions of elliptic equations in a cylinder, and the Maximum Principle for parabolic equations.     

Travelling Wave Solutions in a Tissue Interaction Model for Skin Pattern Formation

We discuss the existence and the uniqueness of travelling wave solutions for a tissue interaction model on skin pattern formation proposed by Cruywagen and Murray. The geometric theory of singular perturbations is employed.     

Electro-elastic interaction between a piezoelectric screw dislocation and circular interfacial rigid lines

The electro-elastic interaction between a piezoelectric screw dislocation located either outside or inside inhomogeneity and circular interfacial rigid lines under anti-plane mechanical and in-plane electrical loads in linear piezoelectric materials is dealt with in the framework of linear elastic theory. Using Riemann–Schwarz’s symmetry principle integrated with the analysis of singularity of complex functions, the general solution of this problem is presented in this paper. For a special example, the closed form solutions for electro-elastic fields in matrix and inhomogeneity regions are derived explicitly when interface containing single rigid line. Applying perturbation technique, perturbation stress and electric displacement fields are obtained. The image force acting on piezoelectric screw dislocation is calculated by using the generalized Peach–Koehler formula. As a result, numerical analysis and discussion show that soft inhomogeneity can repel screw dislocation in piezoelectric material due to their intrinsic electro-mechanical coupling behavior and the influence of interfacial rigid line upon the image force is profound. When the radian of circular rigid line reaches extensive magnitude, the presence of interfacial rigid line can change the interaction mechanism.     

Existence of planar flame fronts in convective-diffusive periodic media

We prove the existence of planar travelling wave solutions in a reaction-diffusion-convection equation with combustion nonlinearity and self-adjoint linear part in R ⁿ, n1. The linear part involves diffusion-convection terms and periodic coefficients. These travelling waves have wrinkled flame fronts propagating with constant effective speeds in periodic inhomogeneous media. We use the method of continuation, spectral theory, and the maximum principle. Uniqueness and monotonicity properties of solutions follow from a previous paper. These properties are essential to overcoming the lack of compactness and the degeneracy in the problem.     

Nonlinear and viscous effects on wave propagation in an elastic axisymmetric vessel

In this paper, a power series and Fourier series approach is used to solve the governing equations of motion in an elastic axisymmetric vessel with the assumption that the fluid is incompressible and Newtonian in a laminar flow. We obtain solutions for the wave speed and attenuation coefficient, analytically where possible, and show how these differ under a number of different conditions. Viscosity is found to reduce the wave speed from that predicted by linear wave theory and the nonlinear terms to increase the wave speed in comparison to the linear solution. For vessels with a wall stiffness in the arterial range, the reduction in the wave speed due to the viscous terms is approximately 10% and the increase due to the nonlinear terms is approximately 5%. This difference between the linear and nonlinear wave speeds was found to be largely constant irrespective of the number of terms considered in the power series for the velocity profile. The linear wave speed was found to vary weakly with stiffness, whilst the nonlinear wave speed was found to vary significantly with the stiffness, especially at low values of stiffness. The 10% variation in the wave speed due to the viscous terms was found to be constant with wall stiffness whilst the 5% variation due to the nonlinear terms was found to vary with wall stiffness. The importance of the number of terms considered in the power series is discussed showing that only a relatively small number is required in the viscous case to obtain accurate results.     

Flow control over an undulating membrane

The flow along a flexible membrane forced to undulate in the form of a streamwise travelling wave pattern is studied experimentally in detail. Flow field and force measurements confirm that the form drag of the wavy wall is significantly reduced when starting the undulatory motion. A mechanical model of an undulating membrane was built, based on previous investigations described in literature, and placed in an open water channel. The motion pattern of the membrane was prescribed in such a way to achieve a downstream travelling wave with increasing amplitude. The exploratory focus laid on the identification of hydrodynamic mechanisms of drag reduction due to undulatory motion. The wave-speed c of the travelling wave was set proportional to the incoming flow velocity U, according to an optimum ratio identified by previous numerical and experimental investigations. Poisson’s equation for the pressure was used to calculate the 2D pressure field from the experimental data of the unsteady flow field. In addition, the integral drag force of the membrane, as a function of c/U, was measured with a force balance to compare with previous published numerical findings. Furthermore, the velocities close to the surface of the membrane were measured, and the boundary layer profiles were determined. The resulting normalised velocity profiles affirm an oscillation between laminar and turbulent flow over one period of the motion. The results are in good agreement with previous experimental and numerical findings. Additionally, the characteristics of the flow along a travelling wave with increasing amplitude are discussed in more detail.     

A wave propagation problem for non-uniform screw dislocation motion in a viscoelastic half-plane

The wave propagation problem for a largely arbitrary anti-plane displacement discontinuity imposed along a line perpendicular to the surface of a stress-free linearly viscoelastic half-plane is considered. The general Laplace transform solution is obtained and then inverted for the case of a screw dislocation moving at an arbitrary speed in a Maxwell material. It is shown that the material viscoelasticity alters the coefficient of the dislocation edge stress singularity and damps the surface displacements from the elastic values. The surface damping increases with time, distance from the dislocation path and dislocation speed, whether sub- or supersonic.     

Homogenization of scalar wave equations with hysteresis

The paper deals with a scalar wave equation of the form where is a Prandtl–Ishlinskii operator and are given functions. This equation describes longitudinal vibrations of an elastoplastic rod. The mass density and the Prandtl–Ishlinskii distribution function are allowed to depend on the space variable x. We prove existence, uniqueness and regularity of solution to a corresponding initial-boundary value problem. The system is then homogenized by considering a sequence of equations of the above type with spatially periodic data and , where the spatial period tends to 0. We identify the homogenized limits and and prove the convergence of solutions to the solution of the homogenized equation. Received June 17, 1999     

MOVING SCREW DISLOCATION IN CUBIC QUASICRYSTAL

The elasticity theory of the dislocation of cubic quasicrystals is developed. The governing equations of anti-plane elasticity dynamics problem of the quasicrystals were reduced to a solution of wave equations by introducing displacement functions, and the analytical expressions of displacements, stresses and energies induced by a moving screw dislocation in the cubic quasicrystalline and the velocity limit of the dislocation were obtained. These provide important information for studying the plastic deformation of the new solid material.     

Stability of travelling fronts in a model for flame propagation Part II: Nonlinear stability

This paper is concerned with the nonlinear stability of the travelling-wave solutions of the multidimensional thermodiffusive model for flame propagation, with unit Lewis number. The model consists in a semilinear parabolic equation in an infinite cylinder, with Neumann boundary conditions. We prove that any solution which is initially close to a travelling wave will converge to a translate of that wave.     

Bifurcations of travelling wave solutions for Jaulent-Miodek equations

By using the theory of bifurcations of planar dynamic systems to the coupled Jaulent-Miodek equations,the existence of smooth solitary travelling wave solutions and uncountably infinite many smooth periodic travelling wave solutions is studied and the bifurcation parametric sets are shown.Under the given parametric conditions,all possible representations of explicit exact solitary wave solutions and periodic wave solutions are obtained.