On axisymmetric traveling waves and radial solutions of semi-linear elliptic equations
Combining analytical techniques from perturbation methods and dynamical systems theory, we present an elementaryapproach to the detailed construction of axisymmetric diffusive interfaces in semi-linear elliptic equations. Solutions of the resulting non-autonomous radial differential equations can be expressed in terms of a slowlyvarying 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 front solutions. These axisym-metric solutions are fundamental examples of more general curved fronts that arise in a wide variety of scientific fields, and we extensivelydiscuss 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. Manyof the results contained in this article are known, and in presenting known results, it is intended that this article be expositoryin nature, providing elementarydemonstrations 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 radiallysymmetric solutions of semilinear elliptic problems, especiallyto those articles in which more advanced mathematical theoryis developed. © 2000 Rocky Mountain Mathematics Consortium.
Witelski, TP; Ono, K; Kaper, TJ
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