Tools from equivariant bifurcation theory are applied to the problem of the optimisation of horizontal planar arrays to minimise array noise gain in a 2D isotropic noise field. We specifically use the performance measure derived in the paper by Hayward [3], which is invariant under suitable actions of the symmetry group ${\bf S}_n\times{\bf O}(2)$, although we would expect most suitable measures to inherit such a symmetry due to the physical properties of the array. An analysis of bifurcations in the presence of this symmetry provides a list of array configurations which one would expect to (locally) optimise performance. This analysis provides a systematic way to search through $2n$-dimensional phase-space, via low dimensional searches, for solutions. We provide an example of the technique for an array with seven hydrophones and a maximum aperture size of $3m$.

It is a great pleasure for me to be asked to write this valedictory message to mark the end of my founding editorship of the European Journal of Applied Mathematics. I am tempted to follow in the discursive footsteps of one of the great founding editors, G. K. Batchelor, of the Journal of Fluid Mechanics fame. However, in accordance with our abiding efforts to keep EJAM crisp and concise, I will simply give you an ‘executive summary’ of my thoughts on what makes an applied mathematics journal successful:

There must be an integration of different styles. Applied mathematics means different things to different people around the world, and it is all too easy for one part of the culture to ignore another on the grounds of being too theoretical, too numerical or too practical. Editors must try to break down these barriers, none more than the board of EJAM, where we have always tried to give equal weight to theory and application. But even (or especially?) within Europe, this balance sometimes appears as hard to achieve as agreement about the euro.

All submissions must be read carefully by at least one editor, in addition to the referees. This is the surest way of minimizing the chances of mistaken rejection (or acceptance) and it is often vital when issues of ‘style’ arise. The reading can be painful for all concerned and I will long remember the following splendid riposte from an anguished author “I see you have eaten the referee's poisoned worm: bon appetit!”.

Along with the personal touch, there must be just the right degree of continuity with which authors can readily identify. Any editorial board striving to promote a particular intellectual policy will invariably have a degree of commonality of interests, and hence a continuity of theme. But this brings the danger of tunnel vision. To balance this, there is a need for a regular and imaginative turnover of board members, and this is an activity with which Cambridge University Press has been most helpful over the years.

Last, but by no means least, is the mandatory requirement that there be an understanding and unfailingly cheerful editorial assistant. This is another respect in which CUP's enlightened attitude has benefited several of their journals, and our authors and editors alike have strong reasons to be thankful to Angela Howard. We are delighted that she will stay on for a few years at least.

The linearized Ginzburg–Landau equations in both semi-infinite strips and rectangles are transformed into equivalent one-dimensional integral equations. Then, the properties of the integral equations are utilized to prove that the onset field for a semi-infinite strip is isolated. We solve the integral equations numerically to obtain the onset field for both rectangles and semi-infinite strips. A formal asymptotic expansion of the onset field in the long rectangle limit is also obtained. Using this formal expansion, we show that the onset field converges in this limit faster than any finite exponential rate, and as a byproduct, that the onset mode in a semi-infinite strip must be asymptotically symmetric.

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.

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.

Under certain conditions the motion of superconducting vortices is primarily governed by an instability. We consider an averaged model, for this phenomenon, describing the motion of large numbers of such vortices. The model equations are parabolic, and, in one spatial dimension $x$, take the form \begin{eqnarray*} {H_2}_t &=& \frac{\partial}{\partial x} ( | {H_3} {H_2}_x-{H_2} {H_3}_x | {H_2}_x ), \\ {H_3}_t &=& \frac{\partial}{\partial x} ( | {H_3} {H_2}_x-{H_2} {H_3}_x | {H_3}_x ). \end{eqnarray*} where $H_2$ and $H_3$ are components of the magnetic field in the $y$ and $z$ directions respectively. These equations have an extremely rich group of symmetries and a correspondingly large class of similarity reductions. In this work, we look for non-trivial steady solutions to the model, deduce their stability and use a numerical method to calculate time-dependent solutions. We then apply Lie Group based similarity methods to calculate a complete catalogue of the model's similarity reductions and use this to investigate a number of its physically important similarity solutions. These describe the short time response of the superconductor as a current or magnetic field is switched on (or off).

In the limit of small activator diffusivity, the stability of a one-spike solution to the shadow Gierer–Meinhardt activator-inhibitor system is studied for various ranges of the reaction-time constant $\tau$ associated with the inhibitor field dynamics. By analyzing the spectrum of the eigenvalue problem associated with the linearization around a one-spike solution, it is proved, for a certain parameter regime, that a complex conjugate pair of eigenvalues crosses into the unstable right half-plane $Re(\lambda) > 0$ as $\tau$ increases past a critical value $\tau_0$. For this parameter regime, it is proved that there are exactly two eigenvalues in the right half-plane when $\tau > \tau_0$ and none when $0 \leq \tau < \tau_0$. It is shown numerically that this critical value of $\tau$ represents the onset of an oscillatory instability in the height of the spike. For other parameter regimes, a similar Hopf bifurcation is confirmed numerically. Full numerical solutions to the shadow problem are computed for a spike that is initially centred at the origin of a radially symmetric domain. Different types of large-scale oscillatory motions for the height of a spike are observed numerically for values of $\tau$ well beyond $\tau_0$.

We consider periodic solutions which bifurcate from equilibria in simple population models which incorporate a state-dependent time delay of the discrete kind. The delay is a function of the current size of the population. Solutions near equilibria are constructed using perturbation methods to determine the sub/supercriticality of the bifurcation and hence their stability. The stability of the bifurcating solutions depends on the qualitative form of the delay function. This is in contrast to the stability of an equilibrium, which is determined purely by the actual value of this function at the equilibrium.

The eddy-current problem for the time-harmonic Maxwell equations in domains and with conductors of general topology is considered. The existence of a unique magnetic field is proved for a suitable weak formulation. An equivalent strong formulation is then derived, where the conditions related to the specific geometry of the domain are made explicit. In particular, a new condition that must be satisfied by the magnetic field on the interface between a multiply-connected conductor and the non-conducting region is determined. Finally, the strong formulation of the problem for the electric field in the non-conducting region is derived, and the existence of a unique solution is proved. In conclusion, this leads to the determination of the complete set of equations describing the eddy-current problem in terms of the magnetic and the electric fields. Whether some commonly-used formulations satisfy the additional condition on the interface is also checked.

We consider heat transfer processes where radiation in a large number of frequency bands plays a dominant role. In the simulation of such processes, the radiative transfer equation has to be solved repeatedly. To obtain an efficient and accurate solution method, we propose a new hybrid algorithm which combines fast solvers for the radiative transfer equation in the low and high absorption regime, respectively. A key role is played by an exact, residual based error formula. The algorithm is applied to a cooling problem of high quality optical glass.

We prove the existence of a solution to a free boundary value problem arising in elastohydrodynamic lubrication. Previously, the existence was proved only under severe restrictions on the range of the physical parameters. We remove those restrictions here.

The equations for three-wave interaction describe the resonant, quadratic, nonlinear interaction of three waves. They are obtained as amplitude equations in an asymptotic reduction of the basic equations of nonlinear optics, fluid mechanics, and plasma physics. These equations are completely integrable and have been the subject of intensive research in the last years. It is the purpose of this paper to prove exact estimates between the approximations obtained via this system and solutions of the original physical system. Although the three-wave interaction model is believed to describe a number of different physical models we restrict attention to its application as a model of the resonant interaction of water waves subject to weak surface tension.

In this paper, we study a class of linear functional differential equations of which the pantograph equation is a prominent member. Specifically, we study the existence of solutions holomorphic at a fixed point of the functional argument. The local theory for equations with attracting fixed points is known [17, 13], but little is known about the case where the fixed point is repelling. We formulate an eigenvalue problem for the repelling fixed point case and show that the corresponding spectrum is discrete. Hence, that holomorphic solutions occur only as special cases. The second order pantograph equation is used to illustrate this result. A key step in this process is to reformulate the problem in terms of a compact operator. Aside from exploiting well known results for the spectra of such operators, we use results such as the Fredholm Alternative to derive existence results for the non-homogeneous problem.

When individuals move together in large groups, as seen in schools of fish, they adapt their speed and direction to that of their neighbours. We present and analyse a model for the speed adaptation process in the case in which all individuals move in the same or in two opposite directions. The model consists of a hyperbolic conservation law for the density of individuals coupled to a parabolic or elliptic equation for speed. A detailed linear analysis reveals several mechanisms for the appearance of instabilities of the homogeneous steady state, which trigger the formation of schools, herds, flocks, etc. Long-term existence of weak solutions is shown using the vanishing viscosity approach.

We present an exact residual based error formula in natural norms for a class of transport equations. The derivation of the error formula relies on an abstract formulation in a general Hilbert space setting. The key role is played by the validity of an inversion formula. Its verification is for particular radiative transfer equations equivalent to the identification of strong and weak traces. The residual based error formula can be used in the design of efficient and accurate simulations of the cooling process of high quality glass [5].

Our task is the identification of the reluctivity $\nu\,{=}\,\nu(B)$ in $\vec{H}\,{=}\,\nu(B) \vec{B}$, ($B\,{=}\,|\vec{B}|$) from measurements of the magnetic flux for different excitation currents in a driving coil, in the context of a nonuniform magnetic field distribution. This is a nonlinear inverse problem and ill-posed in the sense of unstable data dependence, whose solution is done numerically by a Newton type iterative scheme, regularized by an appropriate stopping criterion. The computational complexity of this method is determined by the number of necessary forward evaluations, i.e. the number of numerical solutions to the three-dimensional magnetic field problem. We keep the effort minimal by applying a special discretization strategy to the inverse problem, based on multigrid methods for ill-posed problems. Numerical results demonstrate the efficiency of the proposed method.

Two nonlinear diffusion equations for thin film epitaxy, with or without slope selection, are studied in this work. The nonlinearity models the Ehrlich–Schwoebel effect – the kinetic asymmetry in attachment and detachment of adatoms to and from terrace boundaries. Both perturbation analysis and numerical simulation are presented to show that such an atomistic effect is the origin of a nonlinear morphological instability, in a rough-smooth-rough pattern, that has been experimentally observed as transient in an early stage of epitaxial growth on rough surfaces. Initial-boundary-value problems for both equations are proven to be well-posed, and the solution regularity is also obtained. Galerkin spectral approximations are studied to provide both a priori bounds for proving the well-posedness and numerical schemes for simulation. Numerical results are presented to confirm part of the analysis and to explore the difference between the two models on coarsening dynamics.

The equations describing the pulsating output of a laser containing a saturable absorber are investigated numerically and analytically. The laser admits a singular Hopf bifurcation to a nearly vertical branch of periodic solutions. Using asymptotic methods, we determine a simplified problem that describes the transition from harmonic to pulsating oscillations as the bifurcation parameter is changed. This transition occurs in a layer bounded by the Hopf bifurcation point, and by a critical point near which the branch of solutions becomes vertical.

Calculations, based on exact solutions, of the viscous sintering of simple packings with unimodal and bimodal distributions of particle sizes with shrinking pores are performed. The case of square unit cells is considered in detail. It is found that, for the most part, pore shrinkage times have very weak dependence on the precise details of the pore shape and that accurate estimates of total pore shrinkage times can be obtained based only on knowledge of the initial relative density (green density) and unit cell size. An exception is found when the particle packing is loose, the enclosed pore having a large perimeter-to-area ratio. By studying two different square packings of equal-sized particles it is shown that the looser packing can exhibit dilation, rather than densification, in the early stage of sintering.

We describe various blow-up patterns for the fourth-order one-dimensional semilinear parabolic equation \[ u_t = - u_{xxxx} + \b [(u_x)^3]_x + e^{u} \] with a parameter $\b \ge 0$, which is a model equation from explosion-convection theory. Unlike the classical Frank-Kamenetskii equation $u_t=u_{xx} +e^u$ (a solid fuel model), by using analytical and numerical evidence, we show that the generic blow-up in this fourth-order problem is described by a similarity solution $u_*(x,t) = -\ln(T\,{-}\,t) \,{+}\, f_1(x/(T\,{-}\,t)^{1/4})(T>0$ is the blow-up time), with a non-trivial profile $f_1 \not \equiv 0$. Numerical solution of the PDE shows convergence to the self-similar solution with the profile $f_1$ from a wide variety of initial data. We also construct a countable subset of other, not self-similar, blow-up patterns by using a spectral analysis of an associated linearized operator and matching with similarity solutions of a first-order Hamilton–Jacobi equation.