Rational solutions and rational-oscillatory solutions of the defocusing nonlinear Schrödinger equation are expressed in terms of special polynomials associated with rational solutions of the fourth Painlevé equation. The roots of these special polynomials have a regular, symmetric structure in the complex plane. The rational solutions verify results of Nakamura and Hirota [J. Phys. Soc. Japan, 54 (1985) 491–499] whilst the rational-oscillatory solutions appear to be new solutions of the defocusing nonlinear Schrödinger equation.

In the small diffusion limit ε→0, metastable dynamics is studied for the generalized Burgers problem

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Here u=u(x, t) and f(u) is smooth, convex, and satisfies f(0)=f′(0)=0. The choice f(u)=u2/2 has been shown previously to arise in connection with the physical problem of upward flame-front propagation in a vertical channel in a particular parameter regime. In this context, the shape y=y(x, t) of the flame-front interface satisfies u=−yx. For this problem, it is shown that the principal eigenvalue associated with the linearization around an equilibrium solution corresponding to a parabolic-shaped flame-front interface is exponentially small. This exponentially small eigenvalue then leads to a metastable behaviour for the time- dependent problem. This behaviour is studied quantitatively by deriving an asymptotic ordinary differential equation characterizing the slow motion of the tip location of a parabolic-shaped interface. Similar metastability results are obtained for more general f(u). These asymptotic results are shown to compare very favourably with full numerical computations.

This paper concerns the drainage of a thin liquid lamella into a Plateau border. Many models for draining soap films assume that their gas-liquid interfaces are effectively immobile. Such models predict a phenomenon known as “marginal pinching”, in which the film tends to pinch in the margin between the lamella and the Plateau border. We analyse the opposite extreme, in which the gas-liquid interfaces are assumed to be stress-free. We apply a nonlocal coordinate transformation that transforms the nonlinear governing equations into the linear heat equation. We thus obtain a travelling-wave solution and show that it is globally attractive. We therefore find that no marginal pinching occurs in this case: the film always approaches a monotonic profile at large times. This behaviour reflects a marked qualitative difference between a film with immobile interfaces and one with perfectly mobile interfaces. In the former case, suction of fluid into the Plateau border is localised, while in the latter it is instantaneously transmitted throughout the film.

In this paper, we argue that quantitative empirical research to explain and predict criminal and related behaviour can benefit greatly from explicit theories of action linking individual and contextual factors in the causation of crime. Such theories foster a systematic selection of causal variables for data collection and hypothesis testing instead of a more indiscriminate accumulation of ‘risk factor’ correlates. Moreover, action theory encourages statistical modelling of crime causation beyond the most common linear regression. This paper illustrates both points by estimating two empirical models – a conventional logistic model and a Rasch model – on scenario response data concerning youth violence. The findings of this study show that the extent to which young people indicate a violent response to a provocation is dependent on their (law relevant) morality and ability to exercise self-control as well as the deterrent qualities (monitoring) of the setting.

We study the propagation properties of the reaction-diffusion equation of Fisher type

with p<1<m. Taking into account that solutions of the Cauchy problem are nonunique if m+p⩾2, we prove that the minimal solutions in this case tend to propagate with certain minimal speed c∗(m, p). More precisely, if we translate any solution with a velocity ∣c∣<c∗, we get the limit in time value one, and if the initial value u(·, 0) vanishes, say, for x⩾0, the minimal solution translated with velocity c>c∗ tends to zero. Also, an interface appears for the minimal solutions, whose asymptotic velocity is c∗. This behaviour depends upon the existence of special solutions of travelling wave form. Travelling waves have been widely studied for diffusion equations related with the above. We characterize here the minimal velocity c∗ for which travelling waves exist, as an analytic function of the parameters m and p, for every m+p⩾2, by viewing it as an anomalous exponent. Some local properties of the minimal solutions and their interfaces in the case m+p⩾2 are also proved.

Using mathematical methods to understand and model crime is a recent idea that has drawn considerable attention from researchers during the last five years. From the plethora of models that have been proposed, perhaps the most successful one has been a diffusion-type differential equations model that describes how the number of criminals evolves in a specific area. We propose a more detailed form of this model that allows for two distinct criminal types associated with major and minor crime. Additionally, we examine a stochastic variant of the model that represents more realistically the ‘generation’ of new criminals. Numerical solutions from both models are presented and compared with actual crime data for the Greater Manchester area. Agreement between simulations and actual data is satisfactory. A preliminary statistical analysis of the data also supports the model's potential to describe crime.

We consider a mathematical model which describes the frictional contact between a piezoelectric body and a foundation. The material behaviour is modelled with a non-linear electro-elastic constitutive law, the contact is bilateral, the process is static and the foundation is assumed to be electrically conductive. Both the friction law and the electrical conductivity condition on the contact surface are described with subdifferential boundary conditions. We derive a variational formulation of the problem which is of the form of a system of two coupled hemi-variational inequalities for the displacement and the electric potential fields, respectively. Then we prove the existence of a weak solution to the model and, under additional assumptions, its uniqueness. The proof is based on an abstract result on operator inclusions in Banach spaces.

We derive a class of Navier–Stokes–Cahn–Hilliard systems that models two-phase flows with mass transfer coupled to the process of chemotaxis. These thermodynamically consistent models can be seen as the natural Navier–Stokes analogues of earlier Cahn–Hilliard–Darcy models proposed for modelling tumour growth, and are derived based on a volume-averaged velocity, which yields simpler expressions compared to models derived based on a mass-averaged velocity. Then, we perform mathematical analysis on a simplified model variant with zero excess of total mass and equal densities. We establish the existence of global weak solutions in two and three dimensions for prescribed mass transfer terms. Under additional assumptions, we prove the global strong well-posedness in two dimensions with variable fluid viscosity and mobilities, which also includes a continuous dependence on initial data and mass transfer terms for the chemical potential and the order parameter in strong norms.

The problem of extrapolating asymptotic perturbation-theory expansions in powers of a small variable to large values of the variable tending to infinity is investigated. The analysis is based on self-similar approximation theory. Several types of self-similar approximants are considered and their use in different problems of applied mathematics is illustrated. Self-similar approximants are shown to constitute a powerful tool for extrapolating asymptotic expansions of different natures.

In this paper, we explore some of the various issues that may occur in attempting to fit a dynamical systems (either agent- or continuum-based) model of urban crime to data on just the attack times and locations. We show how one may carry out a regression analysis for the model described by Short et al. (2008, Math. Mod. Meth. Appl. Sci.) by using simulated attack data from the agent-based model. It is discussed how one can incorporate the attack data into the partial differential equations for the expected attractiveness to burgle and the criminal density to predict crime rates between attacks. Using this predicted crime rate, we derive a likelihood function that one can maximise in order to fit parameters and/or initial conditions for the model. We focus on carrying out data assimilation for two different parameter regions, namely in the case where stationary and non-stationary crime hotspots form. It is found that the likelihood function is ‘flat’ for large ranges of parameters, and that this has major implications for crime forecasting. Hence, we look at how one might carry out a goodness-of-fit and forecasting analysis for crime rates given the range of parameter fits. We show how one can use the Kolmogorov–Smirnov statistic to assess the goodness-of-fit. The dynamical systems analysis of the partial differential equations proves invaluable to understanding how the crime rate forecasts depend on the parameters and their sensitivity. Finally, we outline several interesting directions for future research in this area where we believe that the combination of dynamical systems modelling, analysis, and data assimilation can prove effective in developing policing strategies for urban crime.

We undertake the numerical analysis of a reaction-diffusion system of ‘$\lambda$–$\omega$’ type [26]. Results are presented for a fully-practical piecewise linear finite element method by mimicking results in the continuous case [11]. We establish a priori estimates and error bounds for a semi-discrete and a fully discrete finite element approximation. The theoretical results are illustrated and verified via the numerical solution of periodic plane waves in one space dimension. Experiments in two space dimensions led to‘target patterns’ and spiral wave break-up.

Let , N ≥2 be a bounded smooth domain and α > 1. We are interested in the singular elliptic equation with Neumann boundary conditions. In this paper, a complete description of all continuous radially symmetric solutions is given. In particular, we construct nontrivial smooth solutions as well as rupture solutions. Here a continuous solution is said to be a rupture solution if its zero set is nonempty. When N = 2 and α = 3, the equation is used to model steady states of van der Waals force driven thin films of viscous fluids. We also consider the physical problem when total volume of the fluid is prescribed.

Consider a Hele-Shaw cell that is initially empty, and inject fluid at a number of injection points into the gap. To begin with, the plan view of the region occupied by the fluid will consist of growing circular discs, but these will then coalesce and, in general, lead to a multiply-connected geometry. Assuming a constant pressure condition to be relevant at the free boundaries, we show that the entire motion can be explicitly described analytically. When the connectivity is greater than two, the geometry is characterized by a conformal map given by a function that is automorphic with respect to a Schottky group, and we show how to construct this as a ratio of Poincaré theta series. The efficacy of our solution procedure is demonstrated by a number of examples chosen to illustrate points of both physical and mathematical interest.

Dense, dry granular avalanches are very efficient at sorting the larger particles towards the free surface of the flow, and finer grains towards the base, through the combined processes of kinetic sieving and squeeze expulsion. This generates an inversely graded particle-size distribution, which is fundamental to a variety of pattern formation mechanisms, as well as subtle size-mobility feedback effects, leading to the formation of coarse-grained lateral levees that create channels in geophysical flows, enhancing their run-out. In this paper we investigate some of the properties of a recent model [Gray, J. M. N. T. & Thornton, A. R. (2005) A theory for particle size segregation in shallow granular free-surface flows. Proc. R. Soc. 461, 1447–1473]; [Thornton, A. R., Gray, J. M. N. T. & Hogg, A. J. (2006) A three-phase mixture theory for particle size segregation in shallow granular free-surface flows. J. Fluid. Mech. 550, 1–25] for the segregation of particles of two sizes but the same density in a shear flow typical of shallow avalanches. The model is a scalar conservation law in space and time, for the volume fraction of smaller particles, with non-constant coefficients depending on depth within the avalanche. It is proved that for steady flow from an inlet, complete segregation occurs beyond a certain finite distance down the slope, no matter what the mixture at the inlet. In time-dependent flow, dynamic shock waves can develop; they are interfaces separating different mixes of particles. Shock waves are shown to be stable if and only if there is a greater concentration of large particles above the interface than below. Constructions with shocks and rarefaction waves are demonstrated on a pair of physically relevant initial boundary value problems, in which a region of all small particles is penetrated from the inlet by either a uniform mixture of particles or by a layer of small particles over a layer of large particles. In both cases, and under a linear shear flow, solutions are constructed for all time and shown to have similar structure for all choices of parameters.

In the limit of small activator-diffusivity $\varepsilon$, a formal asymptotic analysis is used to derive a differential equation for the motion of a one-spike solution to a simplified form of the Gierer–Meinhardt activator-inhibitor model in a two-dimensional domain. The analysis, which is valid for any finite value of the inhibitor diffusivity $D$ with $D\,{\gg}\,\varepsilon^2$, is delicate in that two disparate scales $\varepsilon$ and ${-1/\ln\varepsilon}$ must be treated. This spike motion is found to depend on the regular part of a reduced-wave Green's function and its gradient. Limiting cases of the dynamics are analyzed. For $D$ small with $\varepsilon^2 \,{\ll}\, D \,{\ll}\, 1$, the spike motion is metastable. For $D\,{\gg}\, 1$, the motion now depends on the gradient of a modified Green's function for the Laplacian. The effect of the shape of the domain and of the value of $D$ on the possible equilibrium positions of a one-spike solution is also analyzed. For $D\,{\ll}\,1$, stable spike-layer locations correspond asymptotically to the centres of the largest radii disks that can be inserted into the domain. Thus, for a dumbbell-shaped domain when $D\,{\ll}\,1$, there are two stable equilibrium positions near the centres of the lobes of the dumbbell. In contrast, for the range $D\,{\gg}\,1$, a complex function method is used to derive an explicit formula for the gradient of the modified Green's function. For a specific dumbbell-shaped domain, this formula is used to show that there is only one equilibrium spike-layer location when $D\,{\gg}\,1$, and it is located in the neck of the dumbbell. Numerical results for other non-convex domains computed from a boundary integral method lead to a similar conclusion regarding the uniqueness of the equilibrium spike location when $D\,{\gg}\,1$. This leads to the conjecture that, when $D\,{\gg}\, 1$, there is only one equilibrium spike-layer location for any convex or non-convex simply connected domain. Finally, the asymptotic results for the spike dynamics are compared with corresponding full numerical results computed using a moving finite element method.

The asymptotic phase θ of an initial point x in the stable manifold of a limit cycle (LC) identifies the phase of the point on the LC to which the flow φt(x) converges as t → ∞. The infinitesimal phase response curve (iPRC) quantifies the change in timing due to a small perturbation of a LC trajectory. For a stable LC in a smooth dynamical system, the iPRC is the gradient ∇x(θ) of the phase function, which can be obtained via the adjoint of the variational equation. For systems with discontinuous dynamics, the standard approach to obtaining the iPRC fails. We derive a formula for the iPRCs of LCs occurring in piecewise smooth (Filippov) dynamical systems of arbitrary dimension, subject to a transverse flow condition. Discontinuous jumps in the iPRC can occur at the boundaries separating subdomains, and are captured by a linear matching condition. The matching matrix, M, can be derived from the saltation matrix arising in the associated variational problem. For the special case of linear dynamics away from switching boundaries, we obtain an explicit expression for the iPRC. We present examples from cell biology (Glass networks) and neuroscience (central pattern generator models). We apply the iPRCs obtained to study synchronization and phase-locking in piecewise smooth LC systems in which synchronization arises solely due to the crossing of switching manifolds.

Many moving boundary problems that are driven in some way by the curvature of the free boundary are gradient flows for the area of the moving interface. Examples are the Mullins–Sekerka flow, the Hele-Shaw flow, flow by mean curvature, and flow by averaged mean curvature. The gradient flow structure suggests an implicit finite differences approach to compute numerical solutions. The proposed numerical scheme will allow us to treat such free boundary problems in both IR2 and IR3. The advantage of such an approach is the re-usability of much of the setup for all of the different problems. As an example of the method, we compute solutions to the averaged mean curvature flow that exhibit the formation of a singularity.

We consider the Gierer–Meinhardt system with precursor inhomogeneity and two small diffusivities in an interval

$$\begin{equation*} \left\{ \begin{array}{ll} A_t=\epsilon^2 A''- \mu(x) A+\frac{A^2}{H}, &x\in(-1, 1),\,t>0,\\[3mm] \tau H_t=D H'' -H+ A^2, & x\in (-1, 1),\,t>0,\\[3mm] A' (-1)= A' (1)= H' (-1) = H' (1) =0, \end{array} \right. \end{equation*}$$

$$\begin{equation*}\mbox{where } \quad 0<\epsilon \ll\sqrt{D}\ll 1, \quad \end{equation*}$$

$$\begin{equation*} \tau\geq 0 \mbox{ and $\tau$ is independent of $\epsilon$. } \end{equation*}$$

$O(\sqrt{D})$

This paper deals with a dynamical inverse problem for a composite beam formed by two connected beams. The vibrations of the composite beam are governed by a differential system where a coupling takes place between longitudinal and bending motions. In this paper, we neglect bending motions and we only deal with the longitudinal motions. These motions are governed by a two-by-two second order system coupled in the lower order terms by the shearing stiffness coefficient, which models the connection between the two beams and which contains direct information on the integrity of the system. We prove that the shearing stiffness coefficient can be reconstructed from the frequency response function of the system evaluated at one end of the beam.

Fibre drawing is an important industrial process for synthetic polymers and optical communications. In the manufacture of optical fibres, precise diameter control is critical to waveguide performance, with tolerances in the submicron range that are met through feedback controls on processing conditions. Fluctuations arise from material non-uniformity plague synthetic polymers but not optical silicate fibres which are drawn from a pristine source. The steady drawing process for glass fibres is well-understood (e.g. [11, 12, 20]). The linearized stability of steady solutions, which characterize limits on draw speed versus processing and material properties, is well-understood (e.g. [9, 10, 11]). Feedback is inherently transient, whereby one adjusts processing conditions in real time based on observations of diameter variations. Our goal in this paper is to delineate the degree of sensitivity to transient fluctuations in processing boundary conditions, for thermal glass fibre steady states that are linearly stable. This is the relevant information for identifying potential sources of observed diameter fluctuation, and for designing the boundary controls necessary to alter existing diameter variations. To evaluate the time-dependent final diameter response to boundary fluctuations, we numerically solve the model nonlinear partial differential equations of thermal glass fibre processing. Our model simulations indicate a relative insensitivity to mechanical effects (such as take-up rates, feed-in rates), but strong sensitivity to thermal fluctuations, which typically form a basis for feedback control.