J. McLeod

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.

Smoothing of Stokes Discontinuities

When the asymptotics of an analytic function f(z) are given in terms of two asymptotic forms S1(z), S2(z), alternately dominant in alternating sectors of the z-plane, Stokes observed that the coefficients of S1, S2, while constant in any one sector, can change from sector to sector, and that to a first approximation the change appears to be discontinuous. Recently, Berry has given an interesting formal argument which shows in more detail how the change takes place, and the present paper gives a rigorous treatment of Berry’s result.

Integral equations and exact solutions for the fourth Painlevé equation

We consider a special case of the fourth Painlevé equation given by d2ƞ / dξ2 = 3ƞ5 + 2ξƞ3 + (1/4ξ2 - v - 1/2 )ƞ, (1) with v a parameter, and seek solutions ƞ(ξ;v) satisfying the boundary condition ƞ(∞)=0. (2) Equation (1) arises as a symmetry reduction of the derivative nonlinear Schrödinger (DNLS) equation, which is a completely integrable soliton equation solvable by inverse scattering techniques. Solutions of equation (1), satisfying (2), are expressed in terms of the solutions of linear integral equations obtained from the inverse scattering formalism for the DNLS equation. We obtain exact ‘bound state’ solutions of equation (1) for v = n, a positive integer, using the integral equation representation, which decay exponentially as ξ→ ± ∞ and are the first example of such solutions for the Painlevé equations. Additionally, using Bäcklund transformations for the fourth Painlevé equation, we derive a nonlinear recurrence relation (commonly referred to as a Bäcklund transformation in the context of soliton equations) for equation (1) relating ƞ(ξ; v) and ƞ(ξ; v + 1).

Integral Equations and Connection Formulae for the Painlevé Equations

We consider special cases of the second, third and fourth Painleve equations given by

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

\frac{{{\text{d}}^2 \eta }} {{{\text{d}}\xi ^2 }} = 2\eta ^3 + \xi \eta

Generalization of the eigenvalues by contour integrals

(1) ,

Integrability of KleinGordon Equations

\frac{{{\text{d}}^2 \eta }} {{{\text{d}}\xi ^2 }} = 2{\text{e}}^\xi + \sinh \eta