We study a unilateral equilibrium problem for the energy functional of a lipid tubule subject to an external field. These tubules, which constitute many biological systems, may form assemblies when they are brought in contact, and so made to adhere to one another along at interstices. The contact energy is taken to be proportional to the area of contact through a constant, which is called the adhesion potential. This competes against the external field in determining the stability of patterns with flat interstices. Though the equilibrium problem is highly nonlinear, we determine explicitly the stability diagram for the adhesion between tubules. We conclude that the higher the field, the lower the adhesion potential needed to make at interstices energetically favourable, though its critical value depends also on the surface tension of the interface between the tubules and the isotropic fluid around them.

Excitation of surface-plasmon resonances of closely spaced nanometallic structures is a key technique used in nanoplasmonics to control light on subwavelength scales and generate highly confined electric-field hotspots. In this paper, we develop asymptotic approximations in the near-contact limit for the entire set of surface-plasmon modes associated with the prototypical sphere dimer geometry. Starting from the quasi-static plasmonic eigenvalue problem, we employ the method of matched asymptotic expansions between a gap region, where the boundaries are approximately paraboloidal, pole regions within the spheres and close to the gap, and a particle-scale region where the spheres appear to touch at leading order. For those modes that are strongly localised to the gap, relating the gap and pole regions gives a set of effective eigenvalue problems formulated over a half space representing one of the poles. We solve these problems using integral transforms, finding asymptotic approximations, singular in the dimensionless gap width, for the eigenvalues and eigenfunctions. In the special case of modes that are both axisymmetric and odd about the plane bisecting the gap, where matching with the outer region introduces a logarithmic dependence upon the dimensionless gap width, our analysis follows Schnitzer [Singular perturbations approach to localized surface-plasmon resonance: nearly touching metal nanospheres. Phys. Rev. B92(23), 235428 (2015)]. We also analyse the so-called anomalous family of even modes, characterised by field distributions excluded from the gap. We demonstrate excellent agreement between our asymptotic formulae and exact calculations.

We give a new characterisation of integrability of a planar vector field at the origin. This allows us to prove that the analytic systems where h, K, Ψ and ξ are analytic functions defined in the neighbourhood of O with K(O) ≠ 0 or Ψ(O) ≠ 0 and n ≥ 1, have a local analytic first integral at the origin. We show new families of analytically integrable systems that are held in the above class. In particular, this class includes all the nilpotent and generalised nilpotent integrable centres that we know.

For superconductors of type II the phenomenon of vortex pinning plays an important role in technological applications. Several models have been proposed for this effect (Kim et al., 1963; Bean, 1964; Bossavit, 1994). In Du et al. (1999) and Prigozhin (1996), some of these models are analyzed. In this work we want to contribute to the analysis for the two-dimensional, rate-independent model proposed in Chapman (2000), which has the special feature that vortex movement and creation is an activated process occurring only when a threshold value of the magnetic field is reached. For analytical studies of related rate-dependent models we refer to Chapman et al. (1996), Schätzle & Styles (1999) and Elliott & Styles (2000).

We investigate the Fano resonance in grating structures using coupled resonators. The grating consists of a perfectly conducting slab with periodically arranged subwavelength slit holes, where inside each period, a pair of slits sit very close to each other. The slit holes act as resonators and are strongly coupled. It is shown rigorously that there exist two groups of resonances corresponding to poles of the scattering problem. One sequence of resonances has imaginary part in the order of ε, where ε is the size of the slit aperture, while the other sequence has imaginary part in the order of ε2. When coupled with the incident wave at resonant frequencies, the narrow-band resonant scattering induced by the latter will interfere with the broader background resonant radiation induced by the former. The interference of these two resonances generates the Fano-type transmission anomaly, which persists in the whole radiation continuum of the grating structure as long as the slit aperture size is small compared to the incident wavelength.

A system of differential equations describing the joint motion of thermo-elastic porous body and slightly compressible viscous thermofluid occupying pore space is considered. Although the problem is correct in an appropriate functional space, it is very hard to tackle due to the fact that its main differential equations involve non-smooth oscillatory coefficients, both big and small, under the differentiation operators. The rigorous justification under various conditions imposed on physical parameters is fulfilled for homogenization procedures as the dimensionless size of the pores tends to zero, while the porous body is geometrically periodic. As a result, we derive Biot's system of equations of thermo-poroelasticity, a similar system, consisting of anisotropic Lamé equations for a thermoelastic solid coupled with acoustic equations for a thermofluid, Darcy's system of filtration, or acoustic equations for a thermofluid, according to ratios between physical parameters. The proofs are based on Nguetseng's two-scale convergence method of homogenization in periodic structures.

We study a system of two second-order differential equations with cubic nonlinearities which model a film of superconductor material subjected to a tangential magnetic field. We verify some recent conjectures of one of the authors about multiplicity of solutions. We show that for an appropriate range of parameter values the relevant boundary value problem has at least two symmetric solutions. It is also proved that a second range of parameters exists for which there are three symmetric solutions.

We consider an initial boundary value problem for the non-local equation, ut = uxx+λf(u)/(∫1-1f (u)dx)2, with Robin boundary conditions. It is known that there exists a critical value of the parameter λ, say λ*, such that for λ > λ* there is no stationary solution and the solution u(x, t) blows up globally in finite time t*, while for λ < λ* there exist stationary solutions. We find, for decreasing f and for λ > λ*, upper and lower bounds for t*, by using comparison methods. For f(u) = e−u, we give an asymptotic estimate: t* ∼ tu(λ−λ*)−1/2 for 0 < (λ−λ*) [Lt ] 1, where tu is a constant. A numerical estimate is obtained using a Crank-Nicolson scheme.

Building on work of Prandtl and Alexander, we study logarithmic vortex spiral solutions of the two-dimensional incompressible Euler equations. We consider multi-branched spirals that are not symmetric, including mixtures of sheets and continuum vorticity. We find that non-trivial solutions allow only sheets, that there is a large variety of such solutions, but that only the Alexander spirals with three or more symmetric branches appear to yield convergent Biot–Savart integral.

In this paper, we present a distributed Proportional-Integral (PI) strategy with self-tuning adaptive gains for reaching asymptotic consensus in networks of non-identical linear agents under constant disturbances. Alternative adaptive strategies are presented, based on global or local measures of the agents' disagreement. The proposed approaches are validated on a representative numerical example. Preliminary analytical results further confirm the viability of the self-tuning strategies.

We study the semi-empirical b—ε model which describes the time evolution of turbulent spots in the case of equal diffusivity of the turbulent energy density b and the energy dissipation rate ε. We prove that the system of two partial differential equations possesses a solution, and that after some time this solution exhibits self-similar behaviour, provided that the system has self-similar solutions. The existence of such self-similar solutions depends upon the value of a parameter of the model.

We consider an optimal control problem based on the evolutionary game theory model introduced by Short et al. (Short, M. B., Brantingham, P. J. & D'Orsogna, M. R. (2010) Cooperation and punishment in an adversarial game: How defectors pave the way to a peaceful society. Phys. Rev. E82(6), 066114.1–066114.7) to study societal attitudes in relation to committing and reporting crimes. Since in [26] (Short, M. B., Brantingham, P. J. & D'Orsogna, M. R. Cooperation and punishment in an adversarial game: How defectors pave the way to a peaceful society. Phys. Rev. E82(6), 066114.1–066114.7) it is shown that the presence of criminal informants leads to diminishing crime, in this paper we investigate the active recruitment of informants from the general population via external intervention, albeit at a cost to society. While higher recruitment levels may be the most beneficial in abating crime, these are also more expensive. We thus formulate our optimal control problem to account for finite resources, incurred costs and expected benefits, and determine the most favourable recruitment strategy under given constraints. We consider the cases of targeted and untargeted recruitment, and allow recruitment costs to depend on past cumulative payoffs within a given memory time-frame so that conversion of more successful individuals may be more costly than that of less successful ones. Our optimal control problem is expressed via three control functions subject to a system of delay differential equations, and is numerically solved, analysed and discussed under different settings and in different parameter regimes. We find that the optimal strategy can change drastically and abruptly as parameters and resource constraints vary, and that increased information on individual player strategies leads to only slightly decreased minimal costs.

Motivated by a parameter identification problem for a fourth-order partial differential equation which models the design of car windscreens by the sagging process, we numerically study a parameter identification problem which is of second-order as an equation for the parameter. The error magnitude and structure for a regularized Landweber method for solving this nonlinear inverse problem with interior data depend markedly on the type of this second-order equation.

The shape stability of the reaction interface for reactive flow in a porous medium is investigated. Previous work showed that the Reaction-Infiltration Instability could cause the reaction zone to lose stability when the Peclet number exceeded a critical value. The new feature of this study is to include a velocity-dependent hydrodynamic dispersion. A mathematical model for this phenomenon is given in the form of a moving free-boundary problem. The spectrum of the linearized problem is obtained, and the related analysis and numerical calculations show that the onset of the instability is not eliminated by the new dispersive terms. The details of analysis show that the instability is reduced especially by the transverse dispersion.

We study two coupled reaction-diffusion equations of the $\lambda$–$\omega$ type [11] in $d\,{\le}\,3$ space dimensions, on a convex bounded domain with a $C^2$ boundary. The equations are close to a supercritical Hopf bifurcation in the reaction kinetics and are model equations for oscillatory reaction-diffusion equations. Global existence, uniqueness and continuous dependence on initial data of strong and weak solutions are proved using the classical Faedo-Galerkin method of Lions [15] and compactness arguments. We also present a complete case study for the application of this method to systems of nonlinear reaction-diffusion equations.

In this paper the free boundary problem for groundwater phreatic surface is represented in the form of a variational principle. It is proved that the flow domain Ω that solves the problem is a minimizer of some functional Λ(Ω). Weak solutions are introduced as minimizers of the lower semi-continuous regularization of Λ(⋅). Within this approach the existence of weak solutions is proved for a wide class of input data.

This study investigates the phenomenon of targeted energy transfer (TET) from a linear oscillator to a nonlinear attachment behaving as a nonlinear energy sink for both transient and stochastic excitations. First, the dynamics of the underlying Hamiltonian system under deterministic transient loading is studied. Assuming that the transient dynamics can be partitioned into slow and fast components, the governing equations of motion corresponding to the slow flow dynamics are derived and the behaviour of the system is analysed. Subsequently, the effect of noise on the slow flow dynamics of the system is investigated. The Itô stochastic differential equations for the noisy system are derived and the corresponding Fokker–Planck equations are numerically solved to gain insights into the behaviour of the system on TET. The effects of the system parameters as well as noise intensity on the optimal regime of TET are studied. The analysis reveals that the interaction of nonlinearities and noise enhances the optimal TET regime as predicted in deterministic analysis.

We develop and analyse a discrete, one-dimensional model of cell motility which incorporates the effects of volume filling, cell-to-cell adhesion and chemotaxis. The formal continuum limit of the model is a non-linear generalisation of the parabolic-elliptic Keller–Segel equations, with a diffusivity which can become negative if the adhesion coefficient is large. The consequent ill-posedness results in the appearance of spatial oscillations and the development of plateaus in numerical solutions of the underlying discrete model. A global-existence result is obtained for the continuum equations in the case of favourable parameter values and data, and a steady-state analysis, which, amongst other things, accounts for high-adhesion plateaus, is carried out. For ill-posed cases, a singular Stefan-problem formulation of the continuum limit is written down and solved numerically, and the numerical solutions are compared with those of the original discrete model.

We discuss the question of uniqueness of planar flames for a simple one-step chemical reaction. We show that when the Lewis number is less than unity (i.e. species diffusion is larger than heat diffusion) uniqueness cannot be generally assumed. An example with three flames, two of them being stable, is exhibited. Other related questions, such as sufficient conditions for uniqueness to hold and high activation energy limits, are discussed.

Reduced ordinary differential equation (ODE) models arising from a high-order lubrication system and describing coarsening dynamics of droplets in nanometric polymer film interacting on a hydrophobically coated solid substrate in the presence of large slippage at the liquid/solid interface are analysed. In the limiting case of infinite slip length corresponding in applications to free films, a collision/absorption model then arises and is solved explicitly. The exact coarsening law is derived for it analytically and confirmed numerically. Existence of a threshold for the decay of initial distributions of droplet distances at infinity at which the coarsening rates switch from algebraic to exponential ones is shown.