- Thomas P. Witelski, K. Ono, T. J. Kaper, On Axi-symmetric Traveling waves and Radial solutions of semi-linear elliptic equations,
Natural Resource Modeling 13, 3, 2000, pp. 339-387
(last updated on 2000/07/19)
Combining analytical techniques from perturbation methods and dynamical systems theory, we present an elementary approach to the detailed construction of axi-symmetric diffusive interfaces in semi-linear elliptic equations. Solutions of the resulting non-autonomous radial differential equations can be expressed in terms of a slowly varying phase plane system. Special analytical results for the phase plane system are used to produce closed-form solutions for the asymptotic forms of the curved fronts solutions. These axi-symmetric problems are fundamental examples of more general curved fronts that arise in a wide variety of scientific fields, and we extensively discuss a number of them, with a particular emphasis on connections to geometric models for the motion of interfaces. Related classical results for traveling waves in one-dimensional problems are also reviewed briefly. Many of the results contained in this article are known, and in presenting known results, it is intended that this article be expository in nature, providing elementary demonstrations of some of the central dynamical phenomena and mathematical techniques. It is hoped that the article serves as one possible avenue of entree to the literature on radially symmetric solutions of semilinear elliptic problems, especially to those articles in which more advanced mathematical theory is developed.