A problem for a backward–forward parabolic partial differential equation with a small diffusion coefficient ε is analyzed. The problem arises in describing the stochastic behaviour of a data-handling system with a large number of sources which turn on or off at random. We solve the problem asymptotically for ε small by using the ray method and the boundary layer method.

The large-time behaviour of a large class of solutions to the two-dimensional linear diffusion equation in situations with radial symmetry is governed by the function known as Ramanujan's integral. This is also true when the diffusion coefficient is complex, which corresponds to Schrödinger's equation. We examine the asymptotic expansion of Ramanujan's integral for large values of its argument over the whole complex plane by considering the analytic continuation of Ramanujan's integral to the left half-plane. The resulting expansions are compared to accurate numerical computations of the integral. The large-time behaviour derived from Ramanujan's integral of the solution to the diffusion equation outside a cylinder is not valid far from the domain boundary. A simple method based on matched asymptotic expansions is outlined to calculate the solution at large times and distances: the resulting form of the solution combines the inverse logarithmic decay in time typical of Ramanujan's integral with spatial dependence on the usual similarity variable for the diffusion equation.

In this paper we describe solutions of a one-dimensional model of inviscid fluid jets that develop finite time singularities in a fractal-like manner. We also discuss the extreme sensitivity of the solutions of this problem with respect to small perturbations of the initial data.

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 study plane shear waves in a continuum model for a saturated soil, assuming a hypoplastic flow rule. Following an earlier paper on velocity-controlled waves [1], we now specify a time- periodic stress disturbance on the boundary of a semi-infinite medium. As before, the stress saturates, tending at large times to a position-independent value, away from the boundary. By contrast, the dependence of this asymptotic state on the boundary data is strikingly different: in the previous case, this dependence was wildly discontinuous, while here it is very simple and smooth.

We consider the bifurcating solutions for the Ginzburg–Landau equations when the superconductor is a film of thickness 2d submitted to an external magnetic field. We refine some results obtained earlier [1] on the stability of bifurcating solutions starting from normal solutions. We prove, in particular, the existence of curves d [map ] κ0(d), defined for large d and tending to 2−1/2 when d [map ] +∞ and κ [map ] d1(κ), defined for small κ and tending to √5/2 when κ [map ] 0, which separate the sets of pairs (κ, d) corresponding to different behaviour of the symmetric bifurcating solutions. In this way, we give in particular a complete answer to the question of stability of symmetric bifurcating solutions in the asymptotics ‘κ fixed-d large’ or ‘d fixed-κ small’.

In the preceding paper of this issue of EJAM, Bolley & Helffer [5] add to their extensive set of results on bifurcation in the one-dimensional Ginzburg–Landau model of superconductivity. One of their new results concerns the direction of bifurcation of symmetric solutions. We give another approach to this problem.