Applications of a WKBJ-type ‘ray ansatz’ to obtain asymptotic solutions of the Helmholtz equation in the high-frequency limit are now standard and underpin the construction of ‘geometrical optics’ ray diagrams in many electromagnetic, acoustic and elastic reflection, transmission and other scattering problems. These applications were subsequently extended by Keller to include other types of rays – called ‘diffracted’ rays – to provide an accessible and impressively accurate theory which is relevant in wide-ranging sets of circumstances. Friedlander and Keller then introduced a modified ray ansatz to extend yet further the scope of ray theory and its applicability to certain other classes of diffraction problems (tangential ray incidence upon an obstructing boundary, for instance) and did so by the inclusion of an extra term proportional to a power of the wave number within the exponent of the initial ansatz. Our purpose here is to generalise this further still by the inclusion of several such terms, ordered in a natural sequence in terms of strategically chosen fractional powers of the large wave number, and to derive a systematic sequence of boundary value problems for the coefficient phase functions that arise within this generalised exponent, as well as one for the leading-order amplitude occurring as a pre-exponential factor. One particular choice of fractional power is considered in detail, and waves with specified radially symmetric or planar wavefronts are then analysed, along with a boundary value problem typifying two-dimensional radiation whereby arbitrary phase and amplitude variations are specified on a prescribed boundary curve. This theory is then applied to the scattering of plane and cylindrical waves at curved boundaries with small-scale perturbations to their underlying profile.

Mass migration of cells (via wave motion) plays an important role in many biological processes, particularly chemotaxis. We study the existence of travelling wave solutions for a chemotaxis model on a microscopic scale. The interaction between nutrients and chemoattractants are considered. Unlike previous approaches, we allow for diffusion of substrates, degradation of chemoattractants and cell growth (constant and linear growth rate). We apply asymptotic methods to investigate the behaviour of the solutions when cells are highly sensitive to extracellular signalling. Explicit solutions are demonstrated, and their biological implications are presented. The results presented here extend and generalize known results.

Outflow from a young star might be regarded as approximately equivalent to flow from a point source. If the fluid consists of charged particles, then the magnetic fields produced are governed by Faraday's law. This simple first approximation yields a linear partial differential equation in spherical polar coordinates, and its solution may be represented as the product of a Legendre polynomial with some function of the radial coordinate. This radial function is shown to involve orthogonal polynomials. Their properties are investigated and recurrence formulae for them are derived. Some of the magnetic fields generated by this simple model are illustrated.

In this paper, an initial-value problem for the modified Korteweg-de Vries (mKdV) equation is addressed. Previous numerical simulations of the solution of

\[ u_{t} - 6u^{2} u_{x}+u_{xxx}=0, \quad -\infty<x<\infty, \quad t>0, \]

\[ u(x,0)=\tanh ( Cx ), \quad -\infty<x<\infty, \]

We study steady streaming in a channel between two parallel permeable walls induced by oscillating (in time) injection/suction of a viscous fluid at the walls. We obtain an asymptotic expansion of the solution of the Navier–Stokes equations in the limit when the amplitude of normal displacements of fluid particles near the walls is much smaller than both the width of the channel and the thickness of the Stokes layer. It is shown that the steady part of the flow in this problem is much stronger than the steady flow produced by vibrations of impermeable boundaries. Another interesting feature of this problem is that the direction of the steady flow is opposite to what one would expect if the flow was produced by vibrations of impermeable walls.

Following the derivation of amplitude equations through a new two-time-scale method [O'Malley, R. E., Jr. & Kirkinis, E (2010) A combined renormalization group-multiple scale method for singularly perturbed problems. Stud. Appl. Math. 124, 383–410], we show that a multi-scale method may often be preferable for solving singularly perturbed problems than the method of matched asymptotic expansions. We illustrate this approach with 10 singularly perturbed ordinary and partial differential equations.

Une modélisation du transport de l'humidité dans des plaques minces stratifiées, quiintègre le phénomène de diffusion préférentielle le long des interfaces entre les plis, est proposée dans ce travail. Une loi d'interface surfacique,asymptotiquement équivalente à une très fine couche de fortediffusivité, est tout d'abord établie par des techniquesasymptotiques, puis introduite au niveau desintercouches d'une plaque mince stratifiée. Des développementsasymptotiques selon l'épaisseur de la plaque permettent alors deproposer un modèle bidimensionnel équivalent pour lecomportement hygroscopique de plaque lamifiée dans le cas où les faces supérieure et inférieure de la plaque sont hygroscopiquement isolées. Desapplications numériques viennent enfin valider et illustrer cesmodèles asymptotiques.