This paper concerns the reflection of high-frequency, monochromatic linear waves of wavenumber k(≫ 1) from smooth boundaries which are O(k−1/2) perturbations away from either a specified near-planar boundary or else from a given smooth, two-dimensional curve of general O(1) curvature. For each class of perturbed boundary, we will consider separately plane and cylindrical wave incidence, with general amplitude profiles of each type of incident field. This interfacial perturbation scaling is canonical in the sense that a ray approach requires a modification to the standard WKBJ ‘ray ansatz’ which, in turn, leads to a leading-order amplitude (or ‘transport’) equation which includes an extra term absent in a standard application of the geometrical theory of diffraction. This extra term is unique to this scaling, and the afore-mentioned modification that is required is an application of a generalised type of ray expansion first posed by F. G. Friedlander and J. B. Keller (1955 Commun. Pure Appl. Math.6, 387–394).

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.

We consider slow–fast delayed systems and discuss pulsating periodic solutions, which are characterised by specific properties that (a) the period of the periodic solution is close to the delay, and (b) these solutions are formed close to a bifurcation threshold. Such solutions were previously found in models of mode-locked lasers. Through a case study of population models, this work demonstrates the existence of similar solutions for a rather wide class of delayed systems. The periodic dynamics originates from the Hopf bifurcation on the positive equilibrium. We show that the continuous transformation of the periodic orbit to the pulsating regime is simultaneous with multiple secondary almost resonant Hopf bifurcations, which the equilibrium undergoes over a short interval of parameter values. We derive asymptotic approximations for the pulsating periodic solution and consider scaling of the solution and its period with the small parameter that measures the ratio of the time scales. The role of competition for the realisation of the bifurcation scenario is highlighted.

The selection of Saffman–Taylor fingers by surface tension has been extensively investigated. In this paper, we are concerned with the existence and selection of steadily translating symmetric finger solutions in a Hele–Shaw cell by small but non-zero kinetic undercooling (ε2). We rigorously conclude that for relative finger width λ near one half, symmetric finger solutions exist in the asymptotic limit of undercooling ε2 → 0 if the Stokes multiplier for a relatively simple non-linear differential equation is zero. This Stokes multiplier S depends on the parameter $\alpha \equiv \frac{2 \lambda -1}{(1-\lambda)}\epsilon^{-\frac{4}{3}}$ and earlier calculations have shown this to be zero for a discrete set of values of α. While this result is similar to that obtained previously for Saffman–Taylor fingers by surface tension, the analysis for the problem with kinetic undercooling exhibits a number of subtleties as pointed out by Chapman and King (2003, The selection of Saffman–Taylor fingers by kinetic undercooling, Journal of Engineering Mathematics, 46, 1–32). The main subtlety is the behaviour of the Stokes lines at the finger tip, where the analysis is complicated by non-analyticity of coefficients in the governing equation.

This article is the third one in a series of papers by the authors on vanishing-viscosity solutions to rate-independent damage systems. While in the first two papers (Knees, D. et al. 2013 Math. Models Methods Appl. Sci.23(4), 565–616; Knees, D. et al. 2015 Nonlinear Anal. Real World Appl.24, 126–162) the assumptions on the spatial domain Ω were kept as general as possible (i.e., non-smooth domain with mixed boundary conditions), we assume here that ∂Ω is smooth and that the type of boundary conditions does not change. This smoother setting allows us to derive enhanced regularity spatial properties both for the displacement and damage fields. Thus, we are in a position to work with a stronger solution notion at the level of the viscous approximating system. The vanishing-viscosity analysis then leads us to obtain the existence of a stronger solution concept for the rate-independent limit system. Furthermore, in comparison to [18, 19], in our vanishing-viscosity analysis we do not switch to an artificial arc-length parameterization of the trajectories but we stay with the true physical time. The resulting concept of Balanced Viscosity solution to the rate-independent damage system thus encodes a more explicit characterization of the system behaviour at time discontinuities of the solution.

Most of the results available on the inverse problem of determining loads acting on elastic beams or plates under transverse vibration refer to single beam or single plate. In this paper, we consider the determination of sources in multi-span systems obtained by connecting either two Euler–Bernoulli elastic beams or two rectangular Kirchhoff–Love elastic plates. The material of the structure is assumed to be homogeneous and isotropic. The transverse load is of the form g(t)f(x), where g(t) is a known function of time and f(x) is the unknown term depending on the position variable x. Under slight a priori assumptions, we prove a uniqueness result for f(x) in terms of observations of the dynamic response taken at interior points of the structure in an arbitrary small interval of time. A numerical implementation of the method is included to show the possible application of the results in the practical identification of the source term.

Recent experiments have shown that patterns can emerge in bacterial colonies genetically modified to have a drop in diffusion when population densities (detected via a quorum sensing mechanism) are sufficiently large. We examine one PDE model of this system, and construct its non-constant stationary solutions in the form of an interface for the bacterial density. We derive the equations for the interface motion and demonstrate analytically that such interface solution is stable when the diffusion rate of bacteria is large and the diffusion rate of signalling molecules, Dh, is small. We further demonstrate that increasing Dh induces a Hopf bifurcation, resulting in a loss of stability, which can lead to either complex spatio-temporal dynamics or extinction of pattern altogether. These results are confirmed by numerical simulations.