Stuart Hastings

A nonlinear elliptic equation arising from gauge field theory and cosmology

We study radially symmetric solutions of a nonlinear elliptic partial differential equation in R2 with critical Sobolev growth, i. e. the nonlinearity is of exponential type. This problem arises from a wide variety of important areas in theoretical physics including superconductivity and cosmology. Our results lead to many interesting implications for the physical problems considered. For example, for the self-dual Chern–Simons theory, we are able to conclude that the electric charge, magnetic flux, or energy of a non-topological N-vortex solution may assume any prescribed value above an explicit lower bound. For the Einstein-matter-gauge equations, we find a necessary and sufficient condition for the existence of a self-dual cosmic string solution. Such a condition imposes an obstruction for the winding number of a string in terms of the universal gravitational constant.

Classical methods in ordinary differential equations : with applications to boundary value problems

Introduction An introduction to shooting methods Some boundary value problems for the Painleve transcendents Periodic solutions of a higher order system A linear example Homoclinic orbits of the FitzHugh-Nagumo equations Singular perturbation problems--rigorous matching Asymptotics beyond all orders Some solutions of the Falkner-Skan equation Poiseuille flow: Perturbation and decay Bending of a tapered rod variational methods and shooting Uniqueness and multiplicity Shooting with more parameters Some problems of A. C. Lazer Chaotic motion of a pendulum Layers and spikes in reaction-diffusion equations, I Uniform expansions for a class of second order problems Layers and spikes in reaction-diffusion equations, II Three unsolved problems Bibliography Index

Layers and spikes in non-homogeneous bistable reaction-diffusion equations

We study e 2 u = f(u,x) = Au(1-u) (Φ-u), where A = A(u,x) > 0, Φ = Φ(x) ∈ (0,1), and e > 0 is sufficiently small, on an interval [0,L] with boundary conditions u = 0 at x = 0, L. All solutions with an e independent number of oscillations are analyzed. Existence of complicated patterns of layers and spikes is proved, and their Morse index is determined. It is observed that the results extend to f = A(u,x) (u - Φ-) (u - Φ) (u - Φ + ) with Φ-(x) < Φ(x) < Φ + (x) and also to an infinite interval.

Diffusion Mediated Transport in Multiple State Systems

Intracellular transport in eukarya is attributed to motor proteins that transduce chemical energy into directed mechanical motion. Nanoscale motors like kinesins tow organelles and other cargo on microtubules or filaments, have a role separating the mitotic spindle during the cell cycle, and perform many other functions. The simplest description gives rise to a weakly coupled system of evolution equations. The transport process, to the minds eye, is analogous to a biased coin toss. We describe how this intuition may be confirmed by a careful analysis of the cooperative effect among the conformational changes and potentials present in the equations.

An Elementary Approach to a Model Problem of Lagerstrom

The equation studied is

DIFFUSION MEDIATED TRANSPORT WITH A LOOK AT MOTOR PROTEINS

u^{\prime\prime}+\frac{n-1}{r}u^{\prime}+\varepsilon u\,u^{\prime}+ku^{\prime2}=0

Static spherically symmetric solutions of a Yang–Mills field coupled to a dilaton

, with boundary conditions

Boundary-value problems in the Fanno model for turbulent compressible flow

u\left(1\right)=0

John Bryce McLeod. 23 December 1929 — 20 August 2014

,

Steady-state analysis of a continuum model for super-infection

u\left(\infty\right) =1