The bifurcation of symmetric superconducting solutions from the normal solution is considered for the one-dimensional Ginzburg–Landau equations by the methods of formal asymptotics. The behaviour of the bifurcating branch depends upon the parameters d, the size of the superconducting slab, and κ, the Ginzburg–Landau parameter. It was found numerically by Aftalion & Troy [1] that there are three distinct regions of the (κ, d) plane, labelled S1, S2 and S3, in which there are at most one, two and three symmetric solutions of the Ginzburg–Landau system, respectively. The curve in the (κ, d) plane across which the bifurcation switches from being subcritical to supercritical is identified, which is the boundary between S2 and S1∪S3, and the bifurcation diagram is analysed in its vicinity. The triple point, corresponding to the point at which S1, S2 and S3 meet, is determined, and the bifurcation diagram and the boundaries of S1, S2 and S3 are analysed in its vicinity. The results provide formal evidence for the resolution of some of the conjectures of Aftalion & Troy [1].

The method of classical Lie symmetries, generalised to differential-difference equations by Quispel, Capel and Sahadevan, is applied to the discrete nonlinear telegraph equation. The symmetry reductions thus obtained are compared with analogous results for the continuous telegraph equation. Some of these ‘continuous’ reductions are used to provide initial data for a numerical scheme which attempts to solve the corresponding discrete equation.

In this note we modify and extend the work of Howison & King [12] to describe the situation of flow around a wedge of arbitrary angle in a Hele-Shaw cell. An ingenious complex-variable method due to Polubarinova-Kochina is used to construct an explicit solution to the zero-surface tension problem.

This paper deals with a wave process induced by an acoustic impulse in a vertically inhomogeneous half-space. The solvability of the corresponding direct initial boundary value problem is proved. We also consider a class of one-dimensional inverse problems including, in particular, the classical inverse problem of the scattering theory, the transmission and reflection problems of seismology, etc. It is shown that all these inverse problems are equivalent in a certain sense, and therefore can be solved by identical methods. Theoretical results are illustrated by numerical examples.

In this paper, a complex variable model for Hele-Shaw moving boundary problems with kinetic undercooling regularization is studied. This model consists of an abstract Cauchy–Kovalevsky problem and of a Riemann–Hilbert–Poincaré problem for holomorphic functions. The local existence of holomorphic solutions is shown. Some numerical results for this model complete the paper.

We consider the classic shortest queue problem in the heavy traffic limit. We assume that the second server works slowly and that the service rate of the first server is nearly equal to the arrival rate. Solving for the (asymptotic) joint steady state queue length distribution involves analyzing a backward parabolic partial differential equation, together with appropriate side conditions. We explicitly solve this problem. We thus obtain a two-dimensional approximation for the steady state queue length probabilities.

The structure of periodic solutions to the Ginzburg–Landau equations in R2 is studied in the critical case, when the equations may be reduced to the first-order Bogomolnyi equations. We prove the existence of periodic solutions when the area of the fundamental cell is greater than 4πM, M being the overall order of the vortices within the fundamental cell (the topological invariant). For smaller fundamental cell areas, it is shown that no periodic solution exists. It is then proved that as the boundaries of the fundamental cell go to infinity, the periodic solutions tend to Taubes' arbitrary N-vortex solution.

We present a simple and rigorous mathematical model and efficient numerical algorithm for the three-dimensional thermal stress analysis of composite structures used in high-temperature catalytic combustors. Numerical experiments are carried out for three types of cell geometries. A homogenization algorithm is implemented, and asymptotic formulae are derived for the effective elastic moduli of the periodic structures.

A moving boundary problem for one phase Hele-Shaw flow with surface tension is considered. The fluid domain is unbounded and its boundary has an infinite length, and a finite number of suction (or injection) points are given. This is a mathematical model of the ‘fingering phenomenon’. We prove the existence of a unique solution locally in time.

Volume 9 (1998), pp. 485–506

One of the author's names was misprinted in this article. EPIFIANO G. VIRGA should read EPIFANIO G. VIRGA.

In addition, on page 502, line 9, the argument leading to evaluate g(∞) := limb→+∞g(b) is not conclusive. The passage

‘On the other hand, if it were g(∞)>0, taking the limit as b→+∞ in (9.9), we would get A=πh2, which by (3.11) is violated by all admissible values of A, but the limiting one. Thus we conclude that g(∞)=0.’

should be changed as follows:

‘On the other hand, taking the limit as b→+∞ in (9.9), we would get A=πh2, by which (3.11) is violated by all admissible values of A, but the limiting one. Thus we conclude that, in the limit as b→+∞, the equilibrium configuration must be singular whenever A>πh2, and so g(∞)=0 because, otherwise, by (9.7) there would be asymptotic regular configurations, as soon as h/2kc(τ0+w) is sufficiently small.’

On page 502, line −7, ‘below’ should be replaced with ‘above’ and ‘regular’ with ‘singular’.

We apologise for the errors in the above paper and hope that no inconvenience has been caused to readers.

We model a thin liquid film moving down a slope using the lubrication approximation with a slip condition. The travelling-wave solution is derived for small inclination angle α, using singular perturbation methods, and compared to the numerical solution. For the linear stability analysis we combine numerical methods with the long-wave approximation and find a small but finite critical α* below which the flow remains linearly stable to spanwise perturbations. This is contrasted with the vanishing of the hump of the travelling-wave solution. Finally, the prevailing linear stability of the travelling-wave at small inclination angles is compared with recent related results using a precursor model. Here, though, a strong dependence on the magnitude of the contact angle is found, which we think has not been observed before.

Short-time existence and uniqueness results in Sobolev spaces are proved for Hele-Shaw flow with kinetic undercooling and for Stokes flow without surface tension. In both cases, the flow is driven by arbitrarily distributed sources and sinks in the interior of the liquid domain. The proofs are based on a general approach consisting of the reformulation of the problem as a Cauchy problem for a nonlinear, nonlocal evolution equation on the unit sphere, quasilinearization by equivariance, investigation of the linearization, and Galerkin approximations. In the situation discussed here, the linearized evolution operator is a first-order differential operator, and thus the evolution equation is of hyperbolic type. Finally, a brief survey of the properties of the evolution equations that arise from Hele-Shaw flow and Stokes flow with and without regularization is given.

Volume 9 (1998), pp. 245–259

The last sentence of §3.2 should read as follows:

For n=4, the splitting yields six such linear PDEs from the coefficients of the terms involving Y(6), Y(4), Y(5), Y(5), Y(4)3, Y(4)2 and Y(4).

In the first paragraph of §6, the penultimate sentence should read as follows:

For an nth-order scalar ODE the determining equations are a linear system of PDEs consisting of the adjoint of the determining equation for symmetries of the nth-order ODE and additional equations when n[ges ]2.

We apologise for the errors in the above paper and hope that no inconvenience has been caused to readers.

Following the historical development of the theory of exact solutions for singularity-driven Hele-Shaw flows, this note demonstrates that the problem of two-dimensional viscous sintering preserves quadrature identities. This provides a unified theoretical perspective in which to understand the two separate free boundary problems. The result is established directly from the equations of motion without appeal to conformal mapping theory, although the result underlies the existence of exact conformal mapping solutions. The formulation leads to a concise, closed-form representation of the evolution equations for the parameters in the conformal mapping function. Some examples are given.

We discuss the application of complex variable methods to Hele-Shaw flows and two-dimensional Stokes flows, both with free boundaries. We outline the theory for the former, in the case where surface tension effects at the moving boundary are ignored. We review the application of complex variable methods to Stokes flows both with and without surface tension, and we explore the parallels between the two problems. We give a detailed discussion of conserved quantities for Stokes flows, and relate them to the Schwarz function of the moving boundary and to the Baiocchi transform of the Airy stress function. We compare the results with the corresponding results for Hele-Shaw flows, the principal consequence being that for Hele-Shaw flows the singularities of the Schwarz function are controlled in the physical plane, while for Stokes flow they are controlled in an auxiliary mapping plane. We illustrate the results with the explicit solutions to specific initial value problems. The results shed light on the construction of solutions to Stokes flows with more than one driving singularity, and on the closely related issue of momentum conservation, which is important in Stokes flows, although it does not arise in Hele-Shaw flows. We also discuss blow-up of zero-surface-tension Stokes flows, and consider a class of weak solutions, valid beyond blow-up, which are obtained as the zero-surface-tension limit of flows with positive surface tension.

In this article the complex variable theory of two-dimensional Stokes flow as developed by Richardson [22], and modified by Howison & Richardson [16], is reviewed. The analysis of [16] is extended to a new solution driven by a point sink, which uses a cubic polynomial conformal mapping (with real coefficients) from the unit disk onto the fluid domain. This solution is analysed in the limit of small surface tension. An apparent ‘stability paradox’ (where two equivalent flow geometries are found, one of which is ‘stable’ and the other unstable) is resolved by allowing the coefficients to take complex values.